:: MESFUNC1 semantic presentation
:: deftheorem Def1 defines INT- MESFUNC1:def 1 :
Lemma2:
0 = - 0
;
theorem Th1: :: MESFUNC1:1
theorem Th2: :: MESFUNC1:2
theorem Th3: :: MESFUNC1:3
:: deftheorem Def2 defines RAT_with_denominator MESFUNC1:def 2 :
theorem Th4: :: MESFUNC1:4
theorem Th5: :: MESFUNC1:5
definition
let c
1 be non
empty set ;
let c
2, c
3 be
PartFunc of c
1,
ExtREAL ;
deffunc H
1(
Element of c
1)
-> Element of
ExtREAL =
(c2 . a1) + (c3 . a1);
func c
2 + c
3 -> PartFunc of a
1,
ExtREAL means :
Def3:
:: MESFUNC1:def 3
(
dom a
4 = ((dom a2) /\ (dom a3)) \ (((a2 " {-infty }) /\ (a3 " {+infty })) \/ ((a2 " {+infty }) /\ (a3 " {-infty }))) & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
4 implies a
4 . b
1 = (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 holds
( b2 in dom b1 implies b1 . b2 = (c2 . b2) + (c3 . b2) ) ) )
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL holds
( dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {-infty }) /\ (c3 " {+infty })) \/ ((c2 " {+infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies 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 holds
( b3 in dom b2 implies b2 . b3 = (c2 . b3) + (c3 . b3) ) ) implies b1 = b2 )
deffunc H
2(
Element of c
1)
-> Element of
ExtREAL =
(c2 . a1) - (c3 . a1);
func c
2 - c
3 -> PartFunc of a
1,
ExtREAL means :: MESFUNC1:def 4
(
dom a
4 = ((dom a2) /\ (dom a3)) \ (((a2 " {+infty }) /\ (a3 " {+infty })) \/ ((a2 " {-infty }) /\ (a3 " {-infty }))) & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
4 implies a
4 . b
1 = (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 holds
( b2 in dom b1 implies b1 . b2 = (c2 . b2) - (c3 . b2) ) ) )
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL holds
( dom b1 = ((dom c2) /\ (dom c3)) \ (((c2 " {+infty }) /\ (c3 " {+infty })) \/ ((c2 " {-infty }) /\ (c3 " {-infty }))) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies 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 holds
( b3 in dom b2 implies b2 . b3 = (c2 . b3) - (c3 . b3) ) ) implies b1 = b2 )
deffunc H
3(
Element of c
1)
-> Element of
ExtREAL =
(c2 . a1) * (c3 . a1);
func c
2 (#) c
3 -> PartFunc of a
1,
ExtREAL means :
Def5:
:: MESFUNC1:def 5
(
dom a
4 = (dom a2) /\ (dom a3) & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
4 implies a
4 . b
1 = (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 holds
( b2 in dom b1 implies b1 . b2 = (c2 . b2) * (c3 . b2) ) ) )
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL holds
( dom b1 = (dom c2) /\ (dom c3) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 . b3 = (c2 . b3) * (c3 . b3) ) ) & dom b2 = (dom c2) /\ (dom c3) & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 . b3 = (c2 . b3) * (c3 . b3) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines + MESFUNC1:def 3 :
:: deftheorem Def4 defines - MESFUNC1:def 4 :
:: deftheorem Def5 defines (#) MESFUNC1:def 5 :
:: deftheorem Def6 defines (#) MESFUNC1:def 6 :
theorem Th6: :: MESFUNC1:6
:: deftheorem Def7 defines - MESFUNC1:def 7 :
:: deftheorem Def8 defines 1. MESFUNC1:def 8 :
:: deftheorem Def9 defines / MESFUNC1:def 9 :
theorem Th7: :: MESFUNC1:7
:: deftheorem Def10 defines |. MESFUNC1:def 10 :
theorem Th8: :: MESFUNC1:8
canceled;
theorem Th9: :: MESFUNC1:9
theorem Th10: :: MESFUNC1:10
theorem Th11: :: MESFUNC1:11
for b
1 being
Real holds
ex b
2 being
Nat st b
1 <= b
2
theorem Th12: :: MESFUNC1:12
for b
1 being
Real holds
ex b
2 being
Nat st
- b
2 <= b
1
theorem Th13: :: MESFUNC1:13
for b
1, b
2 being
real number holds
not ( b
1 < b
2 & ( for b
3 being
Nat holds not 1
/ (b3 + 1) < b
2 - b
1 ) )
theorem Th14: :: MESFUNC1:14
for b
1, b
2 being
real number holds
( ( for b
3 being
Nat holds b
1 - (1 / (b3 + 1)) <= b
2 ) implies b
1 <= b
2 )
theorem Th15: :: MESFUNC1:15
theorem Th16: :: MESFUNC1:16
:: deftheorem Def11 defines is_measurable_on MESFUNC1:def 11 :
theorem Th17: :: MESFUNC1:17
definition
let c
1 be non
empty set ;
let c
2 be
PartFunc of c
1,
ExtREAL ;
let c
3 be
R_eal;
func less_dom c
2,c
3 -> Subset of a
1 means :
Def12:
:: MESFUNC1:def 12
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff ( b
1 in dom a
2 & ex b
2 being
R_eal st
( b
2 = a
2 . b
1 & b
2 < a
3 ) ) );
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 ) ) )
uniqueness
for b1, b2 being Subset of c1 holds
( ( 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 ) ) ) ) implies b1 = b2 )
correctness
;
func less_eq_dom c
2,c
3 -> Subset of a
1 means :
Def13:
:: MESFUNC1:def 13
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff ( b
1 in dom a
2 & ex b
2 being
R_eal st
( b
2 = a
2 . b
1 & b
2 <= a
3 ) ) );
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 ) ) )
uniqueness
for b1, b2 being Subset of c1 holds
( ( 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 ) ) ) ) implies b1 = b2 )
correctness
;
func great_dom c
2,c
3 -> Subset of a
1 means :
Def14:
:: MESFUNC1:def 14
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff ( b
1 in dom a
2 & ex b
2 being
R_eal st
( b
2 = a
2 . b
1 & a
3 < b
2 ) ) );
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 ) ) )
uniqueness
for b1, b2 being Subset of c1 holds
( ( 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 ) ) ) ) implies b1 = b2 )
correctness
;
func great_eq_dom c
2,c
3 -> Subset of a
1 means :
Def15:
:: MESFUNC1:def 15
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff ( b
1 in dom a
2 & ex b
2 being
R_eal st
( b
2 = a
2 . b
1 & a
3 <= b
2 ) ) );
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 ) ) )
uniqueness
for b1, b2 being Subset of c1 holds
( ( 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 ) ) ) ) implies b1 = b2 )
correctness
;
func eq_dom c
2,c
3 -> Subset of a
1 means :
Def16:
:: MESFUNC1:def 16
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff ( b
1 in dom a
2 & ex b
2 being
R_eal st
( b
2 = a
2 . b
1 & a
3 = b
2 ) ) );
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 ) ) )
uniqueness
for b1, b2 being Subset of c1 holds
( ( 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 ) ) ) ) implies b1 = b2 )
correctness
;
end;
:: deftheorem Def12 defines less_dom MESFUNC1:def 12 :
:: deftheorem Def13 defines less_eq_dom MESFUNC1:def 13 :
:: deftheorem Def14 defines great_dom MESFUNC1:def 14 :
:: deftheorem Def15 defines great_eq_dom MESFUNC1:def 15 :
:: deftheorem Def16 defines eq_dom MESFUNC1:def 16 :
theorem Th18: :: MESFUNC1:18
theorem Th19: :: MESFUNC1:19
theorem Th20: :: MESFUNC1:20
theorem Th21: :: MESFUNC1:21
theorem Th22: :: MESFUNC1:22
theorem Th23: :: MESFUNC1:23
theorem Th24: :: MESFUNC1:24
theorem Th25: :: MESFUNC1:25
theorem Th26: :: MESFUNC1:26
theorem Th27: :: MESFUNC1:27
theorem Th28: :: MESFUNC1:28
theorem Th29: :: MESFUNC1:29
theorem Th30: :: MESFUNC1:30
:: deftheorem Def17 defines is_measurable_on MESFUNC1:def 17 :
theorem Th31: :: MESFUNC1:31
theorem Th32: :: MESFUNC1:32
theorem Th33: :: MESFUNC1:33
theorem Th34: :: MESFUNC1:34
theorem Th35: :: MESFUNC1:35
theorem Th36: :: MESFUNC1:36
theorem Th37: :: MESFUNC1:37
theorem Th38: :: MESFUNC1:38
theorem Th39: :: MESFUNC1:39
theorem Th40: :: MESFUNC1:40
theorem Th41: :: MESFUNC1:41