:: BORSUK_2 semantic presentation

Lemma30: for r being real number holds
( ( 0 <= r & r <= 1 ) iff r in the carrier of I[01] )
proof end;

scheme :: BORSUK_2:sch 37
s37{ F1() -> non empty set , F2() -> set , F3( set ) -> set , P1[ set ] } :
Card { F3(w) where w is Element of F1() : ( w in F2() & P1[w] ) } <=` Card F2()
proof end;

theorem Th1: :: BORSUK_2:1
for T1, S, T2, T being non empty TopSpace
for f being Function of T1,S
for g being Function of T2,S st T1 is SubSpace of T & T2 is SubSpace of T & ([#] T1) \/ ([#] T2) = [#] T & T1 is compact & T2 is compact & T is_T2 & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )
proof end;

registration
let T be TopStruct ;
cluster id a1 -> continuous open ;
coherence
( id T is open & id T is continuous )
proof end;
end;

registration
let T be TopStruct ;
cluster one-to-one continuous Relation of the carrier of a1,the carrier of a1;
existence
ex b1 being Function of T,T st
( b1 is continuous & b1 is one-to-one )
proof end;
end;

theorem Th2: :: BORSUK_2:2
canceled;

theorem Th3: :: BORSUK_2:3
for S, T being non empty TopSpace
for f being Function of S,T st f is_homeomorphism holds
f " is open
proof end;

registration
cluster -> real Element of the carrier of I[01] ;
coherence
for b1 being Element of I[01] holds b1 is real
proof end;
end;

theorem Th4: :: BORSUK_2:4
for T being non empty TopSpace
for a being Point of T ex f being Function of I[01] ,T st
( f is continuous & f . 0 = a & f . 1 = a )
proof end;

definition
let T be TopStruct ;
let a be Point of T, b be Point of T;
pred c2,c3 are_connected means :Def1: :: BORSUK_2:def 1
ex f being Function of I[01] ,T st
( f is continuous & f . 0 = a & f . 1 = b );
end;

:: deftheorem Def1 defines are_connected BORSUK_2:def 1 :
for T being TopStruct
for a, b being Point of T holds
( a,b are_connected iff ex f being Function of I[01] ,T st
( f is continuous & f . 0 = a & f . 1 = b ) );

definition
let T be non empty TopSpace;
let a be Point of T, b be Point of T;
redefine pred are_connected as c2,c3 are_connected ;
reflexivity
for a being Point of T holds a,a are_connected
proof end;
end;

definition
let T be TopStruct ;
let a be Point of T, b be Point of T;
assume E32: a,b are_connected ;
mode Path of c2,c3 -> Function of I[01] ,a1 means :Def2: :: BORSUK_2:def 2
( it is continuous & it . 0 = a & it . 1 = b );
existence
ex b1 being Function of I[01] ,T st
( b1 is continuous & b1 . 0 = a & b1 . 1 = b )
by , ;
end;

:: deftheorem Def2 defines Path BORSUK_2:def 2 :
for T being TopStruct
for a, b being Point of T st a,b are_connected holds
for b4 being Function of I[01] ,T holds
( b4 is Path of a,b iff ( b4 is continuous & b4 . 0 = a & b4 . 1 = b ) );

registration
let T be non empty TopSpace;
let a be Point of T;
cluster continuous Path of a2,a2;
existence
ex b1 being Path of a,a st b1 is continuous
proof end;
end;

definition
let T be TopStruct ;
attr a1 is arcwise_connected means :Def3: :: BORSUK_2:def 3
for a, b being Point of T holds a,b are_connected ;
correctness
;
end;

:: deftheorem Def3 defines arcwise_connected BORSUK_2:def 3 :
for T being TopStruct holds
( T is arcwise_connected iff for a, b being Point of T holds a,b are_connected );

registration
cluster non empty arcwise_connected TopStruct ;
existence
ex b1 being TopSpace st
( b1 is arcwise_connected & not b1 is empty )
proof end;
end;

definition
let T be arcwise_connected TopStruct ;
let a be Point of T, b be Point of T;
redefine mode Path of c2,c3 -> Function of I[01] ,a1 means :Def4: :: BORSUK_2:def 4
( it is continuous & it . 0 = a & it . 1 = b );
compatibility
for b1 being Function of I[01] ,T holds
( b1 is Path of a,b iff ( b1 is continuous & b1 . 0 = a & b1 . 1 = b ) )
proof end;
end;

:: deftheorem Def4 defines Path BORSUK_2:def 4 :
for T being arcwise_connected TopStruct
for a, b being Point of T
for b4 being Function of I[01] ,T holds
( b4 is Path of a,b iff ( b4 is continuous & b4 . 0 = a & b4 . 1 = b ) );

registration
let T be arcwise_connected TopStruct ;
let a be Point of T, b be Point of T;
cluster -> continuous Path of a2,a3;
coherence
for b1 being Path of a,b holds b1 is continuous
by ;
end;

Lemma81: ( 0 in [.0,1.] & 1 in [.0,1.] )
proof end;

theorem Th5: :: BORSUK_2:5
for GX being non empty TopSpace st GX is arcwise_connected holds
GX is connected
proof end;

registration
cluster non empty arcwise_connected -> non empty connected TopStruct ;
coherence
for b1 being non empty TopSpace st b1 is arcwise_connected holds
b1 is connected
by ;
end;

Lemma85: for G being non empty TopSpace
for w1, w2, w3 being Point of G
for h1, h2 being Function of I[01] ,G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) )
proof end;

definition
let T be non empty TopSpace;
let a be Point of T, b be Point of T, c be Point of T;
let P be Path of a,b;
let Q be Path of b,c;
assume that
E32: a,b are_connected and
E33: b,c are_connected ;
func c5 + c6 -> Path of a2,a4 means :Def5: :: BORSUK_2:def 5
for t being Point of I[01] holds
( ( t <= 1 / 2 implies it . t = P . (2 * t) ) & ( 1 / 2 <= t implies it . t = Q . ((2 * t) - 1) ) );
existence
ex b1 being Path of a,c st
for t being Point of I[01] holds
( ( t <= 1 / 2 implies b1 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b1 . t = Q . ((2 * t) - 1) ) )
proof end;
uniqueness
for b1, b2 being Path of a,c st ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b1 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b1 . t = Q . ((2 * t) - 1) ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b2 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b2 . t = Q . ((2 * t) - 1) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines + BORSUK_2:def 5 :
for T being non empty TopSpace
for a, b, c being Point of T
for P being Path of a,b
for Q being Path of b,c st a,b are_connected & b,c are_connected holds
for b7 being Path of a,c holds
( b7 = P + Q iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies b7 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b7 . t = Q . ((2 * t) - 1) ) ) );

registration
let T be non empty TopSpace;
let a be Point of T;
cluster constant Path of a2,a2;
existence
ex b1 being Path of a,a st b1 is constant
proof end;
end;

theorem Th6: :: BORSUK_2:6
for T being non empty TopSpace
for a being Point of T
for P being constant Path of a,a holds P = I[01] --> a
proof end;

theorem Th7: :: BORSUK_2:7
for T being non empty TopSpace
for a being Point of T
for P being constant Path of a,a holds P + P = P
proof end;

registration
let T be non empty TopSpace;
let a be Point of T;
let P be constant Path of a,a;
cluster a3 + a3 -> constant ;
coherence
P + P is constant
by ;
end;

definition
let T be non empty TopSpace;
let a be Point of T, b be Point of T;
let P be Path of a,b;
assume E32: a,b are_connected ;
func - c4 -> Path of a3,a2 means :Def6: :: BORSUK_2:def 6
for t being Point of I[01] holds it . t = P . (1 - t);
existence
ex b1 being Path of b,a st
for t being Point of I[01] holds b1 . t = P . (1 - t)
proof end;
uniqueness
for b1, b2 being Path of b,a st ( for t being Point of I[01] holds b1 . t = P . (1 - t) ) & ( for t being Point of I[01] holds b2 . t = P . (1 - t) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines - BORSUK_2:def 6 :
for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b st a,b are_connected holds
for b5 being Path of b,a holds
( b5 = - P iff for t being Point of I[01] holds b5 . t = P . (1 - t) );

Lemma145: for r being Real st 0 <= r & r <= 1 holds
( 0 <= 1 - r & 1 - r <= 1 )
proof end;

Lemma146: for r being Real st r in the carrier of I[01] holds
1 - r in the carrier of I[01]
proof end;

theorem Th8: :: BORSUK_2:8
for T being non empty TopSpace
for a being Point of T
for P being constant Path of a,a holds - P = P
proof end;

registration
let T be non empty TopSpace;
let a be Point of T;
let P be constant Path of a,a;
cluster - a3 -> constant ;
coherence
- P is constant
by ;
end;

theorem Th9: :: BORSUK_2:9
for X, Y being non empty TopSpace
for A being Subset-Family of Y
for f being Function of X,Y holds f " (union A) = union (f " A)
proof end;

definition
let S1 be non empty TopSpace, S2 be non empty TopSpace, T1 be non empty TopSpace, T2 be non empty TopSpace;
let f be Function of S1,S2;
let g be Function of T1,T2;
redefine func [: as [:c5,c6:] -> Function of [:a1,a3:],[:a2,a4:];
coherence
[:f,g:] is Function of [:S1,T1:],[:S2,T2:]
proof end;
end;

theorem Th10: :: BORSUK_2:10
for S1, S2, T1, T2 being non empty TopSpace
for f being continuous Function of S1,T1
for g being continuous Function of S2,T2
for P1, P2 being Subset of [:T1,T2:] st P2 in Base-Appr P1 holds
[:f,g:] " P2 is open
proof end;

theorem Th11: :: BORSUK_2:11
for S1, S2, T1, T2 being non empty TopSpace
for f being continuous Function of S1,T1
for g being continuous Function of S2,T2
for P2 being Subset of [:T1,T2:] st P2 is open holds
[:f,g:] " P2 is open
proof end;

theorem Th12: :: BORSUK_2:12
for S1, S2, T1, T2 being non empty TopSpace
for f being continuous Function of S1,T1
for g being continuous Function of S2,T2 holds [:f,g:] is continuous
proof end;

registration
cluster empty -> T_0 TopStruct ;
coherence
for b1 being TopStruct st b1 is empty holds
b1 is discerning
by T_0TOPSP:def 7;
end;

registration
let T1 be non empty discerning TopSpace, T2 be non empty discerning TopSpace;
cluster [:a1,a2:] -> discerning ;
coherence
[:T1,T2:] is discerning
proof end;
end;

theorem Th13: :: BORSUK_2:13
canceled;

theorem Th14: :: BORSUK_2:14
for T1, T2 being non empty TopSpace st T1 is_T1 & T2 is_T1 holds
[:T1,T2:] is_T1
proof end;

registration
let T1 be non empty being_T1 TopSpace, T2 be non empty being_T1 TopSpace;
cluster [:a1,a2:] -> being_T1 ;
coherence
[:T1,T2:] is being_T1
by ;
end;

Lemma171: for T1, T2 being non empty TopSpace st T1 is_T2 & T2 is_T2 holds
[:T1,T2:] is_T2
proof end;

registration
let T1 be non empty being_T2 TopSpace, T2 be non empty being_T2 TopSpace;
cluster [:a1,a2:] -> being_T2 ;
coherence
[:T1,T2:] is being_T2
by ;
end;

registration
cluster I[01] -> compact being_T2 ;
coherence
( I[01] is compact & I[01] is being_T2 )
proof end;
end;

definition
let T be non empty TopStruct ;
let a be Point of T, b be Point of T;
let P be Path of a,b, Q be Path of a,b;
pred c4,c5 are_homotopic means :: BORSUK_2:def 7
ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = P . s & f . s,1 = Q . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) );
symmetry
for P, Q being Path of a,b st ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = P . s & f . s,1 = Q . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) ) holds
ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = Q . s & f . s,1 = P . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) )
proof end;
end;

:: deftheorem Def7 defines are_homotopic BORSUK_2:def 7 :
for T being non empty TopStruct
for a, b being Point of T
for P, Q being Path of a,b holds
( P,Q are_homotopic iff ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = P . s & f . s,1 = Q . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) ) );

theorem Th15: :: BORSUK_2:15
for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b st a,b are_connected holds
P,P are_homotopic
proof end;

definition
let T be non empty arcwise_connected TopSpace;
let a be Point of T, b be Point of T;
let P be Path of a,b, Q be Path of a,b;
redefine pred are_homotopic as c4,c5 are_homotopic ;
reflexivity
for P being Path of a,b holds P,P are_homotopic
proof end;
end;

theorem Th16: :: BORSUK_2:16
for G being non empty TopSpace
for w1, w2, w3 being Point of G
for h1, h2 being Function of I[01] ,G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) ) by ;

theorem Th17: :: BORSUK_2:17
for T being non empty TopSpace
for a, b, c being Point of T
for G1 being Path of a,b
for G2 being Path of b,c st G1 is continuous & G2 is continuous & G1 . 0 = a & G1 . 1 = b & G2 . 0 = b & G2 . 1 = c holds
( G1 + G2 is continuous & (G1 + G2) . 0 = a & (G1 + G2) . 1 = c )
proof end;

registration
let T be non empty TopSpace;
cluster non empty connected compact Element of bool the carrier of a1;
existence
ex b1 being Subset of T st
( not b1 is empty & b1 is compact & b1 is connected )
proof end;
end;

Lemma191: for p1, p2 being Point of I[01] st p1 <= p2 holds
[.p1,p2.] is non empty Subset of I[01]
by BORSUK_1:83, RCOMP_1:15, RCOMP_1:16;

theorem Th18: :: BORSUK_2:18
for T being non empty TopSpace
for a, b being Point of T st ex f being Function of I[01] ,T st
( f is continuous & f . 0 = a & f . 1 = b ) holds
ex g being Function of I[01] ,T st
( g is continuous & g . 0 = b & g . 1 = a )
proof end;

registration
cluster I[01] -> connected compact being_T2 arcwise_connected ;
coherence
I[01] is arcwise_connected
proof end;
end;