:: MESFUNC1 semantic presentation

definition
func INT- -> Subset of REAL means :Def1: :: MESFUNC1:def 1
for b1 being Real holds
( b1 in a1 iff ex b2 being Nat st b1 = - b2 );
existence
ex b1 being Subset of REAL st
for b2 being Real holds
( b2 in b1 iff ex b3 being Nat st b2 = - b3 )
proof end;
uniqueness
for b1, b2 being Subset of REAL st ( for b3 being Real holds
( b3 in b1 iff ex b4 being Nat st b3 = - b4 ) ) & ( for b3 being Real holds
( b3 in b2 iff ex b4 being Nat st b3 = - b4 ) ) holds
b1 = b2
proof end;
correctness
;
end;

:: deftheorem Def1 defines INT- MESFUNC1:def 1 :
for b1 being Subset of REAL holds
( b1 = INT- iff for b2 being Real holds
( b2 in b1 iff ex b3 being Nat st b2 = - b3 ) );

Lemma2: 0 = - 0
;

registration
cluster INT- -> non empty ;
coherence
not INT- is empty
by Def1, Lemma2;
end;

theorem Th1: :: MESFUNC1:1
NAT , INT- are_equipotent
proof end;

theorem Th2: :: MESFUNC1:2
INT = INT- \/ NAT
proof end;

theorem Th3: :: MESFUNC1:3
NAT , INT are_equipotent by Th1, Th2, CARD_4:15, CARD_4:35;

definition
redefine func INT as INT -> Subset of REAL ;
correctness
coherence
INT is Subset of REAL
;
by NUMBERS:15;
end;

definition
let c1 be Nat;
func RAT_with_denominator c1 -> Subset of RAT means :Def2: :: MESFUNC1:def 2
for b1 being Rational holds
( b1 in a2 iff ex b2 being Integer st b1 = b2 / a1 );
existence
ex b1 being Subset of RAT st
for b2 being Rational holds
( b2 in b1 iff ex b3 being Integer st b2 = b3 / c1 )
proof end;
uniqueness
for b1, b2 being Subset of RAT st ( for b3 being Rational holds
( b3 in b1 iff ex b4 being Integer st b3 = b4 / c1 ) ) & ( for b3 being Rational holds
( b3 in b2 iff ex b4 being Integer st b3 = b4 / c1 ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines RAT_with_denominator MESFUNC1:def 2 :
for b1 being Nat
for b2 being Subset of RAT holds
( b2 = RAT_with_denominator b1 iff for b3 being Rational holds
( b3 in b2 iff ex b4 being Integer st b3 = b4 / b1 ) );

registration
let c1 be Nat;
cluster RAT_with_denominator (a1 + 1) -> non empty ;
coherence
not RAT_with_denominator (c1 + 1) is empty
proof end;
end;

theorem Th4: :: MESFUNC1:4
for b1 being Nat holds INT , RAT_with_denominator (b1 + 1) are_equipotent
proof end;

theorem Th5: :: MESFUNC1:5
NAT , RAT are_equipotent
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
let c3 be set ;
redefine func . as c2 . c3 -> R_eal;
coherence
c2 . c3 is R_eal
proof end;
end;

definition
let c1 be non empty set ;
let c2, c3 be PartFunc of c1, ExtREAL ;
deffunc H1( Element of c1) -> Element of ExtREAL = (c2 . a1) + (c3 . a1);
func c2 + c3 -> PartFunc of a1, ExtREAL means :Def3: :: MESFUNC1:def 3
( dom a4 = ((dom a2) /\ (dom a3)) \ (((a2 " {-infty }) /\ (a3 " {+infty })) \/ ((a2 " {+infty }) /\ (a3 " {-infty }))) & ( for b1 being Element of a1 st b1 in dom a4 holds
a4 . b1 = (a2 . b1) + (a3 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {-infty }) /\ (c3 " {+infty })) \/ ((c2 " {+infty }) /\ (c3 " {-infty }))) & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = (c2 . b2) + (c3 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {-infty }) /\ (c3 " {+infty })) \/ ((c2 " {+infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = (c2 . b3) + (c3 . b3) ) & dom b2 = ((dom c2) /\ (dom c3)) \ (((c2 " {-infty }) /\ (c3 " {+infty })) \/ ((c2 " {+infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = (c2 . b3) + (c3 . b3) ) holds
b1 = b2
proof end;
deffunc H2( Element of c1) -> Element of ExtREAL = (c2 . a1) - (c3 . a1);
func c2 - c3 -> PartFunc of a1, ExtREAL means :: MESFUNC1:def 4
( dom a4 = ((dom a2) /\ (dom a3)) \ (((a2 " {+infty }) /\ (a3 " {+infty })) \/ ((a2 " {-infty }) /\ (a3 " {-infty }))) & ( for b1 being Element of a1 st b1 in dom a4 holds
a4 . b1 = (a2 . b1) - (a3 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {+infty }) /\ (c3 " {+infty })) \/ ((c2 " {-infty }) /\ (c3 " {-infty }))) & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = (c2 . b2) - (c3 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {+infty }) /\ (c3 " {+infty })) \/ ((c2 " {-infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = (c2 . b3) - (c3 . b3) ) & dom b2 = ((dom c2) /\ (dom c3)) \ (((c2 " {+infty }) /\ (c3 " {+infty })) \/ ((c2 " {-infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = (c2 . b3) - (c3 . b3) ) holds
b1 = b2
proof end;
deffunc H3( Element of c1) -> Element of ExtREAL = (c2 . a1) * (c3 . a1);
func c2 (#) c3 -> PartFunc of a1, ExtREAL means :Def5: :: MESFUNC1:def 5
( dom a4 = (dom a2) /\ (dom a3) & ( for b1 being Element of a1 st b1 in dom a4 holds
a4 . b1 = (a2 . b1) * (a3 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = (dom c2) /\ (dom c3) & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = (c2 . b2) * (c3 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = (dom c2) /\ (dom c3) & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = (c2 . b3) * (c3 . b3) ) & dom b2 = (dom c2) /\ (dom c3) & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = (c2 . b3) * (c3 . b3) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines + MESFUNC1:def 3 :
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, ExtREAL holds
( b4 = b2 + b3 iff ( dom b4 = ((dom b2) /\ (dom b3)) \ (((b2 " {-infty }) /\ (b3 " {+infty })) \/ ((b2 " {+infty }) /\ (b3 " {-infty }))) & ( for b5 being Element of b1 st b5 in dom b4 holds
b4 . b5 = (b2 . b5) + (b3 . b5) ) ) );

:: deftheorem Def4 defines - MESFUNC1:def 4 :
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, ExtREAL holds
( b4 = b2 - b3 iff ( dom b4 = ((dom b2) /\ (dom b3)) \ (((b2 " {+infty }) /\ (b3 " {+infty })) \/ ((b2 " {-infty }) /\ (b3 " {-infty }))) & ( for b5 being Element of b1 st b5 in dom b4 holds
b4 . b5 = (b2 . b5) - (b3 . b5) ) ) );

:: deftheorem Def5 defines (#) MESFUNC1:def 5 :
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, ExtREAL holds
( b4 = b2 (#) b3 iff ( dom b4 = (dom b2) /\ (dom b3) & ( for b5 being Element of b1 st b5 in dom b4 holds
b4 . b5 = (b2 . b5) * (b3 . b5) ) ) );

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
let c3 be Real;
deffunc H1( Element of c1) -> Element of ExtREAL = (R_EAL c3) * (c2 . a1);
func c3 (#) c2 -> PartFunc of a1, ExtREAL means :Def6: :: MESFUNC1:def 6
( dom a4 = dom a2 & ( for b1 being Element of a1 st b1 in dom a4 holds
a4 . b1 = (R_EAL a3) * (a2 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = dom c2 & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = (R_EAL c3) * (c2 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = dom c2 & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = (R_EAL c3) * (c2 . b3) ) & dom b2 = dom c2 & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = (R_EAL c3) * (c2 . b3) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines (#) MESFUNC1:def 6 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real
for b4 being PartFunc of b1, ExtREAL holds
( b4 = b3 (#) b2 iff ( dom b4 = dom b2 & ( for b5 being Element of b1 st b5 in dom b4 holds
b4 . b5 = (R_EAL b3) * (b2 . b5) ) ) );

theorem Th6: :: MESFUNC1:6
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real st b3 <> 0 holds
for b4 being Element of b1 st b4 in dom (b3 (#) b2) holds
b2 . b4 = ((b3 (#) b2) . b4) / (R_EAL b3)
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
deffunc H1( Element of c1) -> Element of ExtREAL = - (c2 . a1);
func - c2 -> PartFunc of a1, ExtREAL means :: MESFUNC1:def 7
( dom a3 = dom a2 & ( for b1 being Element of a1 st b1 in dom a3 holds
a3 . b1 = - (a2 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = dom c2 & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = - (c2 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = dom c2 & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = - (c2 . b3) ) & dom b2 = dom c2 & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = - (c2 . b3) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines - MESFUNC1:def 7 :
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b3 = - b2 iff ( dom b3 = dom b2 & ( for b4 being Element of b1 st b4 in dom b3 holds
b3 . b4 = - (b2 . b4) ) ) );

definition
func 1. -> R_eal equals :: MESFUNC1:def 8
1;
correctness
coherence
1 is R_eal
;
by XXREAL_0:def 1;
end;

:: deftheorem Def8 defines 1. MESFUNC1:def 8 :
1. = 1;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
let c3 be Real;
deffunc H1( Element of c1) -> Element of ExtREAL = (R_EAL c3) / (c2 . a1);
func c3 / c2 -> PartFunc of a1, ExtREAL means :Def9: :: MESFUNC1:def 9
( dom a4 = (dom a2) \ (a2 " {0. }) & ( for b1 being Element of a1 st b1 in dom a4 holds
a4 . b1 = (R_EAL a3) / (a2 . b1) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = (dom c2) \ (c2 " {0. }) & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = (R_EAL c3) / (c2 . b2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = (dom c2) \ (c2 " {0. }) & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = (R_EAL c3) / (c2 . b3) ) & dom b2 = (dom c2) \ (c2 " {0. }) & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = (R_EAL c3) / (c2 . b3) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines / MESFUNC1:def 9 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real
for b4 being PartFunc of b1, ExtREAL holds
( b4 = b3 / b2 iff ( dom b4 = (dom b2) \ (b2 " {0. }) & ( for b5 being Element of b1 st b5 in dom b4 holds
b4 . b5 = (R_EAL b3) / (b2 . b5) ) ) );

theorem Th7: :: MESFUNC1:7
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( dom (1 / b2) = (dom b2) \ (b2 " {0. }) & ( for b3 being Element of b1 st b3 in dom (1 / b2) holds
(1 / b2) . b3 = 1. / (b2 . b3) ) )
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
deffunc H1( Element of c1) -> Element of ExtREAL = |.(c2 . a1).|;
func |.c2.| -> PartFunc of a1, ExtREAL means :: MESFUNC1:def 10
( dom a3 = dom a2 & ( for b1 being Element of a1 st b1 in dom a3 holds
a3 . b1 = |.(a2 . b1).| ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = dom c2 & ( for b2 being Element of c1 st b2 in dom b1 holds
b1 . b2 = |.(c2 . b2).| ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL st dom b1 = dom c2 & ( for b3 being Element of c1 st b3 in dom b1 holds
b1 . b3 = |.(c2 . b3).| ) & dom b2 = dom c2 & ( for b3 being Element of c1 st b3 in dom b2 holds
b2 . b3 = |.(c2 . b3).| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines |. MESFUNC1:def 10 :
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b3 = |.b2.| iff ( dom b3 = dom b2 & ( for b4 being Element of b1 st b4 in dom b3 holds
b3 . b4 = |.(b2 . b4).| ) ) );

theorem Th8: :: MESFUNC1:8
canceled;

theorem Th9: :: MESFUNC1:9
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds b2 + b3 = b3 + b2
proof end;

theorem Th10: :: MESFUNC1:10
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds b2 (#) b3 = b3 (#) b2
proof end;

definition
let c1 be non empty set ;
let c2, c3 be PartFunc of c1, ExtREAL ;
redefine func + as c2 + c3 -> PartFunc of a1, ExtREAL ;
commutativity
for b1, b2 being PartFunc of c1, ExtREAL holds b1 + b2 = b2 + b1
by Th9;
redefine func (#) as c2 (#) c3 -> PartFunc of a1, ExtREAL ;
commutativity
for b1, b2 being PartFunc of c1, ExtREAL holds b1 (#) b2 = b2 (#) b1
by Th10;
end;

theorem Th11: :: MESFUNC1:11
for b1 being Realex b2 being Nat st b1 <= b2
proof end;

theorem Th12: :: MESFUNC1:12
for b1 being Realex b2 being Nat st - b2 <= b1
proof end;

theorem Th13: :: MESFUNC1:13
for b1, b2 being real number st b1 < b2 holds
ex b3 being Nat st 1 / (b3 + 1) < b2 - b1
proof end;

theorem Th14: :: MESFUNC1:14
for b1, b2 being real number st ( for b3 being Nat holds b1 - (1 / (b3 + 1)) <= b2 ) holds
b1 <= b2
proof end;

theorem Th15: :: MESFUNC1:15
for b1 being R_eal st ( for b2 being Real holds R_EAL b2 < b1 ) holds
b1 = +infty
proof end;

theorem Th16: :: MESFUNC1:16
for b1 being R_eal st ( for b2 being Real holds b1 < R_EAL b2 ) holds
b1 = -infty
proof end;

definition
let c1 be set ;
let c2 be SigmaField of c1;
let c3 be set ;
pred c3 is_measurable_on c2 means :Def11: :: MESFUNC1:def 11
a3 in a2;
end;

:: deftheorem Def11 defines is_measurable_on MESFUNC1:def 11 :
for b1 being set
for b2 being SigmaField of b1
for b3 being set holds
( b3 is_measurable_on b2 iff b3 in b2 );

theorem Th17: :: MESFUNC1:17
for b1, b2 being set
for b3 being SigmaField of b1 holds
( b2 is_measurable_on b3 iff for b4 being sigma_Measure of b3 holds b2 is_measurable b4 )
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
let c3 be R_eal;
func less_dom c2,c3 -> Subset of a1 means :Def12: :: MESFUNC1:def 12
for b1 being Element of a1 holds
( b1 in a4 iff ( b1 in dom a2 & ex b2 being R_eal st
( b2 = a2 . b1 & b2 < a3 ) ) );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff ( b2 in dom c2 & ex b3 being R_eal st
( b3 = c2 . b2 & b3 < c3 ) ) )
proof end;
uniqueness
for b1, b2 being Subset of c1 st ( for b3 being Element of c1 holds
( b3 in b1 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & b4 < c3 ) ) ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & b4 < c3 ) ) ) ) holds
b1 = b2
proof end;
correctness
;
func less_eq_dom c2,c3 -> Subset of a1 means :Def13: :: MESFUNC1:def 13
for b1 being Element of a1 holds
( b1 in a4 iff ( b1 in dom a2 & ex b2 being R_eal st
( b2 = a2 . b1 & b2 <= a3 ) ) );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff ( b2 in dom c2 & ex b3 being R_eal st
( b3 = c2 . b2 & b3 <= c3 ) ) )
proof end;
uniqueness
for b1, b2 being Subset of c1 st ( for b3 being Element of c1 holds
( b3 in b1 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & b4 <= c3 ) ) ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & b4 <= c3 ) ) ) ) holds
b1 = b2
proof end;
correctness
;
func great_dom c2,c3 -> Subset of a1 means :Def14: :: MESFUNC1:def 14
for b1 being Element of a1 holds
( b1 in a4 iff ( b1 in dom a2 & ex b2 being R_eal st
( b2 = a2 . b1 & a3 < b2 ) ) );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff ( b2 in dom c2 & ex b3 being R_eal st
( b3 = c2 . b2 & c3 < b3 ) ) )
proof end;
uniqueness
for b1, b2 being Subset of c1 st ( for b3 being Element of c1 holds
( b3 in b1 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 < b4 ) ) ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 < b4 ) ) ) ) holds
b1 = b2
proof end;
correctness
;
func great_eq_dom c2,c3 -> Subset of a1 means :Def15: :: MESFUNC1:def 15
for b1 being Element of a1 holds
( b1 in a4 iff ( b1 in dom a2 & ex b2 being R_eal st
( b2 = a2 . b1 & a3 <= b2 ) ) );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff ( b2 in dom c2 & ex b3 being R_eal st
( b3 = c2 . b2 & c3 <= b3 ) ) )
proof end;
uniqueness
for b1, b2 being Subset of c1 st ( for b3 being Element of c1 holds
( b3 in b1 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 <= b4 ) ) ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 <= b4 ) ) ) ) holds
b1 = b2
proof end;
correctness
;
func eq_dom c2,c3 -> Subset of a1 means :Def16: :: MESFUNC1:def 16
for b1 being Element of a1 holds
( b1 in a4 iff ( b1 in dom a2 & ex b2 being R_eal st
( b2 = a2 . b1 & a3 = b2 ) ) );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff ( b2 in dom c2 & ex b3 being R_eal st
( b3 = c2 . b2 & c3 = b3 ) ) )
proof end;
uniqueness
for b1, b2 being Subset of c1 st ( for b3 being Element of c1 holds
( b3 in b1 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 = b4 ) ) ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff ( b3 in dom c2 & ex b4 being R_eal st
( b4 = c2 . b3 & c3 = b4 ) ) ) ) holds
b1 = b2
proof end;
correctness
;
end;

:: deftheorem Def12 defines less_dom MESFUNC1:def 12 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal
for b4 being Subset of b1 holds
( b4 = less_dom b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff ( b5 in dom b2 & ex b6 being R_eal st
( b6 = b2 . b5 & b6 < b3 ) ) ) );

:: deftheorem Def13 defines less_eq_dom MESFUNC1:def 13 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal
for b4 being Subset of b1 holds
( b4 = less_eq_dom b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff ( b5 in dom b2 & ex b6 being R_eal st
( b6 = b2 . b5 & b6 <= b3 ) ) ) );

:: deftheorem Def14 defines great_dom MESFUNC1:def 14 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal
for b4 being Subset of b1 holds
( b4 = great_dom b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff ( b5 in dom b2 & ex b6 being R_eal st
( b6 = b2 . b5 & b3 < b6 ) ) ) );

:: deftheorem Def15 defines great_eq_dom MESFUNC1:def 15 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal
for b4 being Subset of b1 holds
( b4 = great_eq_dom b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff ( b5 in dom b2 & ex b6 being R_eal st
( b6 = b2 . b5 & b3 <= b6 ) ) ) );

:: deftheorem Def16 defines eq_dom MESFUNC1:def 16 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal
for b4 being Subset of b1 holds
( b4 = eq_dom b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff ( b5 in dom b2 & ex b6 being R_eal st
( b6 = b2 . b5 & b3 = b6 ) ) ) );

theorem Th18: :: MESFUNC1:18
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set
for b4 being R_eal st b3 c= dom b2 holds
b3 /\ (great_eq_dom b2,b4) = b3 \ (b3 /\ (less_dom b2,b4))
proof end;

theorem Th19: :: MESFUNC1:19
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set
for b4 being R_eal st b3 c= dom b2 holds
b3 /\ (great_dom b2,b4) = b3 \ (b3 /\ (less_eq_dom b2,b4))
proof end;

theorem Th20: :: MESFUNC1:20
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set
for b4 being R_eal st b3 c= dom b2 holds
b3 /\ (less_eq_dom b2,b4) = b3 \ (b3 /\ (great_dom b2,b4))
proof end;

theorem Th21: :: MESFUNC1:21
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set
for b4 being R_eal st b3 c= dom b2 holds
b3 /\ (less_dom b2,b4) = b3 \ (b3 /\ (great_eq_dom b2,b4))
proof end;

theorem Th22: :: MESFUNC1:22
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set
for b4 being R_eal holds b3 /\ (eq_dom b2,b4) = (b3 /\ (great_eq_dom b2,b4)) /\ (less_eq_dom b2,b4)
proof end;

theorem Th23: :: MESFUNC1:23
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set
for b6 being Real st ( for b7 being Nat holds b3 . b7 = b5 /\ (great_dom b4,(R_EAL (b6 - (1 / (b7 + 1))))) ) holds
b5 /\ (great_eq_dom b4,(R_EAL b6)) = meet (rng b3)
proof end;

theorem Th24: :: MESFUNC1:24
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being SigmaField of b1
for b4 being Function of NAT ,b3
for b5 being set
for b6 being real number st ( for b7 being Nat holds b4 . b7 = b5 /\ (less_dom b2,(R_EAL (b6 + (1 / (b7 + 1))))) ) holds
b5 /\ (less_eq_dom b2,(R_EAL b6)) = meet (rng b4)
proof end;

theorem Th25: :: MESFUNC1:25
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being SigmaField of b1
for b4 being Function of NAT ,b3
for b5 being set
for b6 being real number st ( for b7 being Nat holds b4 . b7 = b5 /\ (less_eq_dom b2,(R_EAL (b6 - (1 / (b7 + 1))))) ) holds
b5 /\ (less_dom b2,(R_EAL b6)) = union (rng b4)
proof end;

theorem Th26: :: MESFUNC1:26
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set
for b6 being Real st ( for b7 being Nat holds b3 . b7 = b5 /\ (great_eq_dom b4,(R_EAL (b6 + (1 / (b7 + 1))))) ) holds
b5 /\ (great_dom b4,(R_EAL b6)) = union (rng b3)
proof end;

theorem Th27: :: MESFUNC1:27
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set st ( for b6 being Nat holds b3 . b6 = b5 /\ (great_dom b4,(R_EAL b6)) ) holds
b5 /\ (eq_dom b4,+infty ) = meet (rng b3)
proof end;

theorem Th28: :: MESFUNC1:28
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set st ( for b6 being Nat holds b3 . b6 = b5 /\ (less_dom b4,(R_EAL b6)) ) holds
b5 /\ (less_dom b4,+infty ) = union (rng b3)
proof end;

theorem Th29: :: MESFUNC1:29
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set st ( for b6 being Nat holds b3 . b6 = b5 /\ (less_dom b4,(R_EAL (- b6))) ) holds
b5 /\ (eq_dom b4,-infty ) = meet (rng b3)
proof end;

theorem Th30: :: MESFUNC1:30
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Function of NAT ,b2
for b4 being PartFunc of b1, ExtREAL
for b5 being set st ( for b6 being Nat holds b3 . b6 = b5 /\ (great_dom b4,(R_EAL (- b6))) ) holds
b5 /\ (great_dom b4,-infty ) = union (rng b3)
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be PartFunc of c1, ExtREAL ;
let c4 be Element of c2;
pred c3 is_measurable_on c4 means :Def17: :: MESFUNC1:def 17
for b1 being real number holds a4 /\ (less_dom a3,(R_EAL b1)) is_measurable_on a2;
end;

:: deftheorem Def17 defines is_measurable_on MESFUNC1:def 17 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 holds
( b3 is_measurable_on b4 iff for b5 being real number holds b4 /\ (less_dom b3,(R_EAL b5)) is_measurable_on b2 );

theorem Th31: :: MESFUNC1:31
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 st b4 c= dom b3 holds
( b3 is_measurable_on b4 iff for b5 being real number holds b4 /\ (great_eq_dom b3,(R_EAL b5)) is_measurable_on b2 )
proof end;

theorem Th32: :: MESFUNC1:32
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 holds
( b3 is_measurable_on b4 iff for b5 being real number holds b4 /\ (less_eq_dom b3,(R_EAL b5)) is_measurable_on b2 )
proof end;

theorem Th33: :: MESFUNC1:33
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 st b4 c= dom b3 holds
( b3 is_measurable_on b4 iff for b5 being real number holds b4 /\ (great_dom b3,(R_EAL b5)) is_measurable_on b2 )
proof end;

theorem Th34: :: MESFUNC1:34
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4, b5 being Element of b2 st b5 c= b4 & b3 is_measurable_on b4 holds
b3 is_measurable_on b5
proof end;

theorem Th35: :: MESFUNC1:35
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4, b5 being Element of b2 st b3 is_measurable_on b4 & b3 is_measurable_on b5 holds
b3 is_measurable_on b4 \/ b5
proof end;

theorem Th36: :: MESFUNC1:36
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5, b6 being Real st b3 is_measurable_on b4 & b4 c= dom b3 holds
(b4 /\ (great_dom b3,(R_EAL b5))) /\ (less_dom b3,(R_EAL b6)) is_measurable_on b2
proof end;

theorem Th37: :: MESFUNC1:37
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 st b3 is_measurable_on b4 & b4 c= dom b3 holds
b4 /\ (eq_dom b3,+infty ) is_measurable_on b2
proof end;

theorem Th38: :: MESFUNC1:38
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 st b3 is_measurable_on b4 holds
b4 /\ (eq_dom b3,-infty ) is_measurable_on b2
proof end;

theorem Th39: :: MESFUNC1:39
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2 st b3 is_measurable_on b4 & b4 c= dom b3 holds
(b4 /\ (great_dom b3,-infty )) /\ (less_dom b3,+infty ) is_measurable_on b2
proof end;

theorem Th40: :: MESFUNC1:40
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2
for b6 being Real st b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= dom b4 holds
(b5 /\ (less_dom b3,(R_EAL b6))) /\ (great_dom b4,(R_EAL b6)) is_measurable_on b2
proof end;

theorem Th41: :: MESFUNC1:41
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5 being Real st b3 is_measurable_on b4 & b4 c= dom b3 holds
b5 (#) b3 is_measurable_on b4
proof end;