:: URYSOHN3 semantic presentation

registration
let c1 be non empty Subset of REAL ;
cluster -> real Element of a1;
coherence
for b1 being Element of c1 holds b1 is real
;
end;

Lemma1: for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 <> {} & b2 misses b3 & ( for b4 being Function of dyadic 0, bool the carrier of b1 holds
not ( b2 c= b4 . 0 & b3 = ([#] b1) \ (b4 . 1) & ( for b5, b6 being Element of dyadic 0 holds
( b5 < b6 implies ( b4 . b5 is open & b4 . b6 is open & Cl (b4 . b5) c= b4 . b6 ) ) ) ) ) )
proof end;

theorem Th1: :: URYSOHN3:1
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Nat holds
ex b5 being Function of dyadic b4, bool the carrier of b1 st
( b2 c= b5 . 0 & b3 = ([#] b1) \ (b5 . 1) & ( for b6, b7 being Element of dyadic b4 holds
( b6 < b7 implies ( b5 . b6 is open & b5 . b7 is open & Cl (b5 . b6) c= b5 . b7 ) ) ) ) )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
let c4 be Nat;
assume E3: ( c1 is_T4 & c2 <> {} & c2 is closed & c3 is closed & c2 misses c3 ) ;
mode Drizzle of c2,c3,c4 -> Function of dyadic a4, bool the carrier of a1 means :Def1: :: URYSOHN3:def 1
( a2 c= a5 . 0 & a3 = ([#] a1) \ (a5 . 1) & ( for b1, b2 being Element of dyadic a4 holds
( b1 < b2 implies ( a5 . b1 is open & a5 . b2 is open & Cl (a5 . b1) c= a5 . b2 ) ) ) );
existence
ex b1 being Function of dyadic c4, bool the carrier of c1 st
( c2 c= b1 . 0 & c3 = ([#] c1) \ (b1 . 1) & ( for b2, b3 being Element of dyadic c4 holds
( b2 < b3 implies ( b1 . b2 is open & b1 . b3 is open & Cl (b1 . b2) c= b1 . b3 ) ) ) )
by E3, Th1;
end;

:: deftheorem Def1 defines Drizzle URYSOHN3:def 1 :
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Nat holds
( b1 is_T4 & b2 <> {} & b2 is closed & b3 is closed & b2 misses b3 implies for b5 being Function of dyadic b4, bool the carrier of b1 holds
( b5 is Drizzle of b2,b3,b4 iff ( b2 c= b5 . 0 & b3 = ([#] b1) \ (b5 . 1) & ( for b6, b7 being Element of dyadic b4 holds
( b6 < b7 implies ( b5 . b6 is open & b5 . b7 is open & Cl (b5 . b6) c= b5 . b7 ) ) ) ) ) );

theorem Th2: :: URYSOHN3:2
canceled;

theorem Th3: :: URYSOHN3:3
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Nat
for b5 being Drizzle of b2,b3,b4 holds
ex b6 being Drizzle of b2,b3,b4 + 1 st
for b7 being Element of dyadic (b4 + 1) holds
( b7 in dyadic b4 implies b6 . b7 = b5 . b7 ) )
proof end;

theorem Th4: :: URYSOHN3:4
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Nat
for b5 being Drizzle of b2,b3,b4 holds
b5 is Element of PFuncs DYADIC ,(bool the carrier of b1)
proof end;

definition
let c1, c2 be non empty set ;
let c3 be Function of NAT , PFuncs c1,c2;
let c4 be Nat;
redefine func . as c3 . c4 -> Element of PFuncs a1,a2;
coherence
c3 . c4 is Element of PFuncs c1,c2
by FUNCT_2:7;
end;

theorem Th5: :: URYSOHN3:5
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 <> {} & b2 misses b3 & ( for b4 being Functional_Sequence of DYADIC , bool the carrier of b1 holds
not for b5 being Nat holds
( b4 . b5 is Drizzle of b2,b3,b5 & ( for b6 being Element of dom (b4 . b5) holds (b4 . b5) . b6 = (b4 . (b5 + 1)) . b6 ) ) ) )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
assume E7: ( c1 is_T4 & c2 <> {} & c2 is closed & c3 is closed & c2 misses c3 ) ;
mode Rain of c2,c3 -> Functional_Sequence of DYADIC , bool the carrier of a1 means :Def2: :: URYSOHN3:def 2
for b1 being Nat holds
( a4 . b1 is Drizzle of a2,a3,b1 & ( for b2 being Element of dom (a4 . b1) holds (a4 . b1) . b2 = (a4 . (b1 + 1)) . b2 ) );
existence
ex b1 being Functional_Sequence of DYADIC , bool the carrier of c1 st
for b2 being Nat holds
( b1 . b2 is Drizzle of c2,c3,b2 & ( for b3 being Element of dom (b1 . b2) holds (b1 . b2) . b3 = (b1 . (b2 + 1)) . b3 ) )
by E7, Th5;
end;

:: deftheorem Def2 defines Rain URYSOHN3:def 2 :
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b1 is_T4 & b2 <> {} & b2 is closed & b3 is closed & b2 misses b3 implies for b4 being Functional_Sequence of DYADIC , bool the carrier of b1 holds
( b4 is Rain of b2,b3 iff for b5 being Nat holds
( b4 . b5 is Drizzle of b2,b3,b5 & ( for b6 being Element of dom (b4 . b5) holds (b4 . b5) . b6 = (b4 . (b5 + 1)) . b6 ) ) ) );

definition
let c1 be Real;
assume E8: c1 in DYADIC ;
func inf_number_dyadic c1 -> Nat means :Def3: :: URYSOHN3:def 3
( ( a1 in dyadic 0 implies a2 = 0 ) & ( a2 = 0 implies a1 in dyadic 0 ) & ( for b1 being Nat holds
( a1 in dyadic (b1 + 1) & not a1 in dyadic b1 implies a2 = b1 + 1 ) ) );
existence
ex b1 being Nat st
( ( c1 in dyadic 0 implies b1 = 0 ) & ( b1 = 0 implies c1 in dyadic 0 ) & ( for b2 being Nat holds
( c1 in dyadic (b2 + 1) & not c1 in dyadic b2 implies b1 = b2 + 1 ) ) )
proof end;
uniqueness
for b1, b2 being Nat holds
not ( ( c1 in dyadic 0 implies b1 = 0 ) & ( b1 = 0 implies c1 in dyadic 0 ) & ( for b3 being Nat holds
( c1 in dyadic (b3 + 1) & not c1 in dyadic b3 implies b1 = b3 + 1 ) ) & ( c1 in dyadic 0 implies b2 = 0 ) & ( b2 = 0 implies c1 in dyadic 0 ) & ( for b3 being Nat holds
( c1 in dyadic (b3 + 1) & not c1 in dyadic b3 implies b2 = b3 + 1 ) ) & not b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines inf_number_dyadic URYSOHN3:def 3 :
for b1 being Real holds
( b1 in DYADIC implies for b2 being Nat holds
( b2 = inf_number_dyadic b1 iff ( ( b1 in dyadic 0 implies b2 = 0 ) & ( b2 = 0 implies b1 in dyadic 0 ) & ( for b3 being Nat holds
( b1 in dyadic (b3 + 1) & not b1 in dyadic b3 implies b2 = b3 + 1 ) ) ) ) );

theorem Th6: :: URYSOHN3:6
for b1 being Real holds
( b1 in DYADIC implies b1 in dyadic (inf_number_dyadic b1) )
proof end;

theorem Th7: :: URYSOHN3:7
for b1 being Real holds
( b1 in DYADIC implies for b2 being Nat holds
( inf_number_dyadic b1 <= b2 implies b1 in dyadic b2 ) )
proof end;

theorem Th8: :: URYSOHN3:8
for b1 being Real
for b2 being Nat holds
( b1 in dyadic b2 implies inf_number_dyadic b1 <= b2 )
proof end;

theorem Th9: :: URYSOHN3:9
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Real holds
( b5 in DYADIC implies for b6 being Nat holds (b4 . (inf_number_dyadic b5)) . b5 = (b4 . ((inf_number_dyadic b5) + b6)) . b5 ) )
proof end;

theorem Th10: :: URYSOHN3:10
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Real holds
not ( b5 in DYADIC & ( for b6 being Subset of b1 holds
ex b7 being Nat st
( b5 in dyadic b7 & not b6 = (b4 . b7) . b5 ) ) ) )
proof end;

theorem Th11: :: URYSOHN3:11
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3 holds
ex b5 being Function of DOM , bool the carrier of b1 st
for b6 being Real holds
( b6 in DOM implies ( ( b6 in halfline 0 implies b5 . b6 = {} ) & ( b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1 ) & ( b6 in DYADIC implies for b7 being Nat holds
( b6 in dyadic b7 implies b5 . b6 = (b4 . b7) . b6 ) ) ) ) )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
assume E15: ( c1 is_T4 & c2 <> {} & c2 is closed & c3 is closed & c2 misses c3 ) ;
let c4 be Rain of c2,c3;
func Tempest c4 -> Function of DOM , bool the carrier of a1 means :Def4: :: URYSOHN3:def 4
for b1 being Real holds
( b1 in DOM implies ( ( b1 in halfline 0 implies a5 . b1 = {} ) & ( b1 in right_open_halfline 1 implies a5 . b1 = the carrier of a1 ) & ( b1 in DYADIC implies for b2 being Nat holds
( b1 in dyadic b2 implies a5 . b1 = (a4 . b2) . b1 ) ) ) );
existence
ex b1 being Function of DOM , bool the carrier of c1 st
for b2 being Real holds
( b2 in DOM implies ( ( b2 in halfline 0 implies b1 . b2 = {} ) & ( b2 in right_open_halfline 1 implies b1 . b2 = the carrier of c1 ) & ( b2 in DYADIC implies for b3 being Nat holds
( b2 in dyadic b3 implies b1 . b2 = (c4 . b3) . b2 ) ) ) )
by E15, Th11;
uniqueness
for b1, b2 being Function of DOM , bool the carrier of c1 holds
( ( for b3 being Real holds
( b3 in DOM implies ( ( b3 in halfline 0 implies b1 . b3 = {} ) & ( b3 in right_open_halfline 1 implies b1 . b3 = the carrier of c1 ) & ( b3 in DYADIC implies for b4 being Nat holds
( b3 in dyadic b4 implies b1 . b3 = (c4 . b4) . b3 ) ) ) ) ) & ( for b3 being Real holds
( b3 in DOM implies ( ( b3 in halfline 0 implies b2 . b3 = {} ) & ( b3 in right_open_halfline 1 implies b2 . b3 = the carrier of c1 ) & ( b3 in DYADIC implies for b4 being Nat holds
( b3 in dyadic b4 implies b2 . b3 = (c4 . b4) . b3 ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines Tempest URYSOHN3:def 4 :
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b1 is_T4 & b2 <> {} & b2 is closed & b3 is closed & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Function of DOM , bool the carrier of b1 holds
( b5 = Tempest b4 iff for b6 being Real holds
( b6 in DOM implies ( ( b6 in halfline 0 implies b5 . b6 = {} ) & ( b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1 ) & ( b6 in DYADIC implies for b7 being Nat holds
( b6 in dyadic b7 implies b5 . b6 = (b4 . b7) . b6 ) ) ) ) ) );

definition
let c1 be non empty set ;
let c2 be TopSpace;
let c3 be Function of c1, bool the carrier of c2;
let c4 be Element of c1;
redefine func . as c3 . c4 -> Subset of a2;
correctness
coherence
c3 . c4 is Subset of c2
;
proof end;
end;

theorem Th12: :: URYSOHN3:12
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Real
for b6 being Subset of b1 holds
( b6 = (Tempest b4) . b5 & b5 in DOM implies b6 is open ) )
proof end;

theorem Th13: :: URYSOHN3:13
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5, b6 being Real holds
( b5 in DOM & b6 in DOM & b5 < b6 implies for b7 being Subset of b1 holds
( b7 = (Tempest b4) . b5 implies Cl b7 c= (Tempest b4) . b6 ) ) )
proof end;

theorem Th14: :: URYSOHN3:14
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3
for b5 being Point of b1 holds
ex b6 being Subset of ExtREAL st
for b7 being set holds
( b7 in b6 iff ( b7 in DYADIC & ( for b8 being Real holds
not ( b8 = b7 & b5 in (Tempest b4) . b8 ) ) ) )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
let c4 be Rain of c2,c3;
let c5 be Point of c1;
func Rainbow c5,c4 -> Subset of ExtREAL means :Def5: :: URYSOHN3:def 5
for b1 being set holds
( b1 in a6 iff ( b1 in DYADIC & ( for b2 being Real holds
not ( b2 = b1 & a5 in (Tempest a4) . b2 ) ) ) );
existence
ex b1 being Subset of ExtREAL st
for b2 being set holds
( b2 in b1 iff ( b2 in DYADIC & ( for b3 being Real holds
not ( b3 = b2 & c5 in (Tempest c4) . b3 ) ) ) )
by Th14;
uniqueness
for b1, b2 being Subset of ExtREAL holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in DYADIC & ( for b4 being Real holds
not ( b4 = b3 & c5 in (Tempest c4) . b4 ) ) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in DYADIC & ( for b4 being Real holds
not ( b4 = b3 & c5 in (Tempest c4) . b4 ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def5 defines Rainbow URYSOHN3:def 5 :
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3
for b5 being Point of b1
for b6 being Subset of ExtREAL holds
( b6 = Rainbow b5,b4 iff for b7 being set holds
( b7 in b6 iff ( b7 in DYADIC & ( for b8 being Real holds
not ( b8 = b7 & b5 in (Tempest b4) . b8 ) ) ) ) );

definition
let c1, c2 be non empty TopSpace;
let c3 be Function of c1,c2;
let c4 be Point of c1;
redefine func . as c3 . c4 -> Point of a2;
correctness
coherence
c3 . c4 is Point of c2
;
proof end;
end;

theorem Th15: :: URYSOHN3:15
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3
for b5 being Point of b1 holds Rainbow b5,b4 c= DYADIC
proof end;

theorem Th16: :: URYSOHN3:16
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3 holds
ex b5 being Function of b1,R^1 st
for b6 being Point of b1 holds
( ( Rainbow b6,b4 = {} implies b5 . b6 = 0 ) & ( for b7 being non empty Subset of ExtREAL holds
( b7 = Rainbow b6,b4 implies b5 . b6 = sup b7 ) ) )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
let c4 be Rain of c2,c3;
func Thunder c4 -> Function of a1,R^1 means :Def6: :: URYSOHN3:def 6
for b1 being Point of a1 holds
( ( Rainbow b1,a4 = {} implies a5 . b1 = 0 ) & ( for b2 being non empty Subset of ExtREAL holds
( b2 = Rainbow b1,a4 implies a5 . b1 = sup b2 ) ) );
existence
ex b1 being Function of c1,R^1 st
for b2 being Point of c1 holds
( ( Rainbow b2,c4 = {} implies b1 . b2 = 0 ) & ( for b3 being non empty Subset of ExtREAL holds
( b3 = Rainbow b2,c4 implies b1 . b2 = sup b3 ) ) )
by Th16;
uniqueness
for b1, b2 being Function of c1,R^1 holds
( ( for b3 being Point of c1 holds
( ( Rainbow b3,c4 = {} implies b1 . b3 = 0 ) & ( for b4 being non empty Subset of ExtREAL holds
( b4 = Rainbow b3,c4 implies b1 . b3 = sup b4 ) ) ) ) & ( for b3 being Point of c1 holds
( ( Rainbow b3,c4 = {} implies b2 . b3 = 0 ) & ( for b4 being non empty Subset of ExtREAL holds
( b4 = Rainbow b3,c4 implies b2 . b3 = sup b4 ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def6 defines Thunder URYSOHN3:def 6 :
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3
for b5 being Function of b1,R^1 holds
( b5 = Thunder b4 iff for b6 being Point of b1 holds
( ( Rainbow b6,b4 = {} implies b5 . b6 = 0 ) & ( for b7 being non empty Subset of ExtREAL holds
( b7 = Rainbow b6,b4 implies b5 . b6 = sup b7 ) ) ) );

definition
let c1 be non empty TopSpace;
let c2 be Function of c1,R^1 ;
let c3 be Point of c1;
redefine func . as c2 . c3 -> Real;
correctness
coherence
c2 . c3 is Real
;
proof end;
end;

theorem Th17: :: URYSOHN3:17
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being Rain of b2,b3
for b5 being Point of b1
for b6 being non empty Subset of ExtREAL holds
( b6 = Rainbow b5,b4 implies for b7 being R_eal holds
( b7 = 1 implies ( 0. <= sup b6 & sup b6 <= b7 ) ) )
proof end;

theorem Th18: :: URYSOHN3:18
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Element of DOM
for b6 being Point of b1 holds
( (Thunder b4) . b6 < b5 implies b6 in (Tempest b4) . b5 ) )
proof end;

theorem Th19: :: URYSOHN3:19
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Real holds
( b5 in DYADIC \/ (right_open_halfline 1) & 0 < b5 implies for b6 being Point of b1 holds
( b6 in (Tempest b4) . b5 implies (Thunder b4) . b6 <= b5 ) ) )
proof end;

theorem Th20: :: URYSOHN3:20
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3
for b5 being Element of DOM holds
( 0 < b5 implies for b6 being Point of b1 holds
not ( b5 < (Thunder b4) . b6 & b6 in (Tempest b4) . b5 ) ) )
proof end;

theorem Th21: :: URYSOHN3:21
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
( b2 <> {} & b2 misses b3 implies for b4 being Rain of b2,b3 holds
( Thunder b4 is continuous & ( for b5 being Point of b1 holds
( 0 <= (Thunder b4) . b5 & (Thunder b4) . b5 <= 1 & ( b5 in b2 implies (Thunder b4) . b5 = 0 ) & ( b5 in b3 implies (Thunder b4) . b5 = 1 ) ) ) ) )
proof end;

theorem Th22: :: URYSOHN3:22
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 <> {} & b2 misses b3 & ( for b4 being Function of b1,R^1 holds
not ( b4 is continuous & ( for b5 being Point of b1 holds
( 0 <= b4 . b5 & b4 . b5 <= 1 & ( b5 in b2 implies b4 . b5 = 0 ) & ( b5 in b3 implies b4 . b5 = 1 ) ) ) ) ) )
proof end;

theorem Th23: :: URYSOHN3:23
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 misses b3 & ( for b4 being Function of b1,R^1 holds
not ( b4 is continuous & ( for b5 being Point of b1 holds
( 0 <= b4 . b5 & b4 . b5 <= 1 & ( b5 in b2 implies b4 . b5 = 0 ) & ( b5 in b3 implies b4 . b5 = 1 ) ) ) ) ) )
proof end;

theorem Th24: :: URYSOHN3:24
for b1 being non empty compact being_T2 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 misses b3 & ( for b4 being Function of b1,R^1 holds
not ( b4 is continuous & ( for b5 being Point of b1 holds
( 0 <= b4 . b5 & b4 . b5 <= 1 & ( b5 in b2 implies b4 . b5 = 0 ) & ( b5 in b3 implies b4 . b5 = 1 ) ) ) ) ) ) by Th23;