:: BORSUK_5 semantic presentation

theorem :: BORSUK_5:1
canceled;

theorem Th2: :: BORSUK_5:2
for A, B, a being set st A c= B & B c= A \/ {a} & not A \/ {a} = B holds
A = B
proof end;

theorem :: BORSUK_5:3
for x1, x2, x3, x4, x5, x6 being set holds {x1,x2,x3,x4,x5,x6} = {x1,x3,x6} \/ {x2,x4,x5}
proof end;

definition
let x1, x2, x3, x4, x5, x6 be set ;
pred x1,x2,x3,x4,x5,x6 are_mutually_different means :Def1: :: BORSUK_5:def 1
( x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6 );
end;

:: deftheorem Def1 defines are_mutually_different BORSUK_5:def 1 :
for x1, x2, x3, x4, x5, x6 being set holds
( x1,x2,x3,x4,x5,x6 are_mutually_different iff ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6 ) );

definition
let x1, x2, x3, x4, x5, x6, x7 be set ;
pred x1,x2,x3,x4,x5,x6,x7 are_mutually_different means :Def2: :: BORSUK_5:def 2
( x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7 );
end;

:: deftheorem Def2 defines are_mutually_different BORSUK_5:def 2 :
for x1, x2, x3, x4, x5, x6, x7 being set holds
( x1,x2,x3,x4,x5,x6,x7 are_mutually_different iff ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7 ) );

theorem Th4: :: BORSUK_5:4
for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3,x4,x5,x6 are_mutually_different holds
card {x1,x2,x3,x4,x5,x6} = 6
proof end;

theorem :: BORSUK_5:5
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
card {x1,x2,x3,x4,x5,x6,x7} = 7
proof end;

theorem Th6: :: BORSUK_5:6
for x1, x2, x3, x4, x5, x6 being set st {x1,x2,x3} misses {x4,x5,x6} holds
( x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 )
proof end;

theorem :: BORSUK_5:7
for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3 are_mutually_different & x4,x5,x6 are_mutually_different & {x1,x2,x3} misses {x4,x5,x6} holds
x1,x2,x3,x4,x5,x6 are_mutually_different
proof end;

theorem :: BORSUK_5:8
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6 are_mutually_different & {x1,x2,x3,x4,x5,x6} misses {x7} holds
x1,x2,x3,x4,x5,x6,x7 are_mutually_different
proof end;

theorem :: BORSUK_5:9
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
x7,x1,x2,x3,x4,x5,x6 are_mutually_different
proof end;

theorem :: BORSUK_5:10
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
x1,x2,x5,x3,x6,x7,x4 are_mutually_different
proof end;

Lm1: 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 )
by BORSUK_2:18;

Lm2: R^1 is arcwise_connected
proof end;

registration
cluster R^1 -> arcwise_connected ;
coherence
R^1 is arcwise_connected by Lm2;
end;

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

Lm3: [#] R^1 = REAL
by TOPMETR:24;

notation
let a be real number ;
synonym ].-infty,a.] for left_closed_halfline a;
synonym ].-infty,a.[ for left_open_halfline a;
synonym [.a,+infty.[ for right_closed_halfline a;
synonym ].a,+infty.[ for right_open_halfline a;
end;

theorem :: BORSUK_5:11
canceled;

theorem :: BORSUK_5:12
canceled;

theorem :: BORSUK_5:13
canceled;

theorem Th14: :: BORSUK_5:14
for a, b being real number holds
( a in ].b,+infty.[ iff a > b )
proof end;

theorem Th15: :: BORSUK_5:15
for a, b being real number holds
( a in [.b,+infty.[ iff a >= b )
proof end;

theorem Th16: :: BORSUK_5:16
for a, b being real number holds
( a in ].-infty,b.] iff a <= b )
proof end;

theorem Th17: :: BORSUK_5:17
for a, b being real number holds
( a in ].-infty,b.[ iff a < b )
proof end;

theorem Th18: :: BORSUK_5:18
for a being real number holds REAL \ {a} = ].-infty,a.[ \/ ].a,+infty.[
proof end;

theorem Th19: :: BORSUK_5:19
for a, b, c, d being real number st a < b & b <= c holds
[.a,b.] misses ].c,d.]
proof end;

theorem Th20: :: BORSUK_5:20
for a, b, c, d being real number st a < b & b <= c holds
[.a,b.[ misses [.c,d.]
proof end;

theorem :: BORSUK_5:21
for A, B being Subset of R^1
for a, b, c, d being real number st a < b & b <= c & c < d & A = [.a,b.[ & B = ].c,d.] holds
A,B are_separated
proof end;

Lm4: for a being real number holds REAL \ ].-infty,a.[ = [.a,+infty.[
by LIMFUNC1:24;

Lm5: for a being real number holds REAL \ ].-infty,a.] = ].a,+infty.[
by LIMFUNC1:24;

Lm6: for a being real number holds REAL \ [.a,+infty.[ = ].-infty,a.[
by LIMFUNC1:24;

theorem :: BORSUK_5:22
canceled;

theorem :: BORSUK_5:23
canceled;

theorem :: BORSUK_5:24
canceled;

theorem :: BORSUK_5:25
canceled;

theorem Th26: :: BORSUK_5:26
for a being real number holds ].-infty,a.] misses ].a,+infty.[
proof end;

theorem Th27: :: BORSUK_5:27
for a being real number holds ].-infty,a.[ misses [.a,+infty.[
proof end;

theorem Th28: :: BORSUK_5:28
for a, b, c being real number st a <= c & c <= b holds
[.a,b.] \/ [.c,+infty.[ = [.a,+infty.[
proof end;

theorem Th29: :: BORSUK_5:29
for a, b, c being real number st a <= c & c <= b holds
].-infty,c.] \/ [.a,b.] = ].-infty,b.]
proof end;

registration
cluster -> real Element of RAT ;
coherence
for b1 being Element of RAT holds b1 is real ;
end;

registration
cluster -> real Element of the carrier of RealSpace ;
coherence
for b1 being Element of RealSpace holds b1 is real by METRIC_1:def 14, XREAL_0:def 1;
end;

theorem :: BORSUK_5:30
canceled;

theorem :: BORSUK_5:31
canceled;

theorem :: BORSUK_5:32
canceled;

theorem :: BORSUK_5:33
for A being Subset of R^1
for p being Point of RealSpace holds
( p in Cl A iff for r being real number st r > 0 holds
Ball p,r meets A ) by GOBOARD6:95, TOPMETR:def 7;

theorem Th34: :: BORSUK_5:34
for p, q being Element of RealSpace st q >= p holds
dist p,q = q - p
proof end;

theorem Th35: :: BORSUK_5:35
for A being Subset of R^1 st A = RAT holds
Cl A = the carrier of R^1
proof end;

theorem Th36: :: BORSUK_5:36
for A being Subset of R^1
for a, b being real number st A = ].a,b.[ & a <> b holds
Cl A = [.a,b.]
proof end;

registration
cluster number_e -> irrational ;
coherence
number_e is irrational by IRRAT_1:42;
end;

definition
func IRRAT -> Subset of REAL equals :: BORSUK_5:def 3
REAL \ RAT ;
coherence
REAL \ RAT is Subset of REAL ;
end;

:: deftheorem defines IRRAT BORSUK_5:def 3 :
IRRAT = REAL \ RAT ;

definition
let a, b be real number ;
func RAT a,b -> Subset of REAL equals :: BORSUK_5:def 4
RAT /\ ].a,b.[;
coherence
RAT /\ ].a,b.[ is Subset of REAL ;
func IRRAT a,b -> Subset of REAL equals :: BORSUK_5:def 5
IRRAT /\ ].a,b.[;
coherence
IRRAT /\ ].a,b.[ is Subset of REAL ;
end;

:: deftheorem defines RAT BORSUK_5:def 4 :
for a, b being real number holds RAT a,b = RAT /\ ].a,b.[;

:: deftheorem defines IRRAT BORSUK_5:def 5 :
for a, b being real number holds IRRAT a,b = IRRAT /\ ].a,b.[;

theorem Th37: :: BORSUK_5:37
for x being real number holds
( x is irrational iff x in IRRAT )
proof end;

registration
cluster real irrational set ;
existence
ex b1 being real number st b1 is irrational by IRRAT_1:42;
end;

registration
cluster IRRAT -> non empty ;
coherence
not IRRAT is empty by Th37, IRRAT_1:42;
end;

theorem Th38: :: BORSUK_5:38
for a being rational number
for b being real irrational number holds a + b is irrational
proof end;

theorem Th39: :: BORSUK_5:39
for a being real irrational number holds - a is irrational
proof end;

theorem Th40: :: BORSUK_5:40
for a being rational number
for b being real irrational number holds a - b is irrational
proof end;

theorem Th41: :: BORSUK_5:41
for a being rational number
for b being real irrational number holds b - a is irrational
proof end;

theorem Th42: :: BORSUK_5:42
for a being rational number
for b being real irrational number st a <> 0 holds
a * b is irrational
proof end;

theorem Th43: :: BORSUK_5:43
for a being rational number
for b being real irrational number st a <> 0 holds
b / a is irrational
proof end;

registration
cluster real irrational -> non zero real set ;
coherence
for b1 being real number st b1 is irrational holds
not b1 is empty
proof end;
end;

theorem Th44: :: BORSUK_5:44
for a being rational number
for b being real irrational number st a <> 0 holds
a / b is irrational
proof end;

theorem Th45: :: BORSUK_5:45
for r being real irrational number holds frac r is irrational
proof end;

registration
let r be real irrational number ;
cluster frac r -> non zero irrational ;
coherence
frac r is irrational by Th45;
end;

registration
let a be real irrational number ;
cluster - a -> non zero irrational ;
coherence
- a is irrational by Th39;
end;

registration
let a be rational number ;
let b be real irrational number ;
cluster a + b -> non zero irrational ;
coherence
a + b is irrational by Th38;
cluster b + a -> non zero irrational ;
coherence
b + a is irrational ;
cluster a - b -> non zero irrational ;
coherence
a - b is irrational by Th40;
cluster b - a -> non zero irrational ;
coherence
b - a is irrational by Th41;
end;

registration
cluster non zero rational set ;
existence
not for b1 being rational number holds b1 is empty
proof end;
end;

registration
let a be non zero rational number ;
let b be real irrational number ;
cluster a * b -> non zero irrational ;
coherence
a * b is irrational by Th42;
cluster b * a -> non zero irrational ;
coherence
b * a is irrational ;
cluster a / b -> non zero irrational ;
coherence
a / b is irrational by Th44;
cluster b / a -> non zero irrational ;
coherence
b / a is irrational by Th43;
end;

theorem :: BORSUK_5:46
canceled;

theorem Th47: :: BORSUK_5:47
for a, b being real number st a < b holds
ex p1, p2 being rational number st
( a < p1 & p1 < p2 & p2 < b )
proof end;

Lm7: for s1, s3, s4, l being real number st s1 <= s3 & s1 < s4 & 0 < l & l < 1 holds
s1 < ((1 - l) * s3) + (l * s4)
by XREAL_1:179;

Lm8: for s1, s3, s4, l being real number st s3 < s1 & s4 <= s1 & 0 < l & l < 1 holds
((1 - l) * s3) + (l * s4) < s1
by XREAL_1:180;

theorem :: BORSUK_5:48
canceled;

theorem :: BORSUK_5:49
canceled;

theorem Th50: :: BORSUK_5:50
for a, b being real number st a < b holds
ex p being real irrational number st
( a < p & p < b )
proof end;

theorem Th51: :: BORSUK_5:51
for A being Subset of R^1 st A = IRRAT holds
Cl A = the carrier of R^1
proof end;

Lm9: for A being Subset of R^1
for a, b being real number st a < b & A = RAT a,b holds
( a in Cl A & b in Cl A )
proof end;

Lm10: for A being Subset of R^1
for a, b being real number st a < b & A = IRRAT a,b holds
( a in Cl A & b in Cl A )
proof end;

theorem Th52: :: BORSUK_5:52
for a, b, c being real number st a < b holds
( c in RAT a,b iff ( not c is irrational & a < c & c < b ) )
proof end;

theorem Th53: :: BORSUK_5:53
for a, b, c being real number st a < b holds
( c in IRRAT a,b iff ( c is irrational & a < c & c < b ) )
proof end;

theorem Th54: :: BORSUK_5:54
for A being Subset of R^1
for a, b being real number st a < b & A = RAT a,b holds
Cl A = [.a,b.]
proof end;

theorem Th55: :: BORSUK_5:55
for A being Subset of R^1
for a, b being real number st a < b & A = IRRAT a,b holds
Cl A = [.a,b.]
proof end;

theorem Th56: :: BORSUK_5:56
for T being connected TopSpace
for A being open closed Subset of T holds
( A = {} or A = [#] T )
proof end;

theorem Th57: :: BORSUK_5:57
for A being Subset of R^1 st A is closed & A is open & not A = {} holds
A = REAL
proof end;

theorem Th58: :: BORSUK_5:58
for A being Subset of R^1
for a, b being real number st A = [.a,b.[ & a <> b holds
Cl A = [.a,b.]
proof end;

theorem Th59: :: BORSUK_5:59
for A being Subset of R^1
for a, b being real number st A = ].a,b.] & a <> b holds
Cl A = [.a,b.]
proof end;

theorem :: BORSUK_5:60
for A being Subset of R^1
for a, b, c being real number st A = [.a,b.[ \/ ].b,c.] & a < b & b < c holds
Cl A = [.a,c.]
proof end;

theorem Th61: :: BORSUK_5:61
for A being Subset of R^1
for a being real number st A = {a} holds
Cl A = {a}
proof end;

theorem Th62: :: BORSUK_5:62
for A being Subset of REAL
for B being Subset of R^1 st A = B holds
( A is open iff B is open )
proof end;

Lm11: for a being real number holds ].a,+infty.[ is open
by FCONT_3:7;

Lm12: for a being real number holds ].-infty,a.] is closed
by FCONT_3:6;

Lm13: for a being real number holds ].-infty,a.[ is open
by FCONT_3:8;

Lm14: for a being real number holds [.a,+infty.[ is closed
by FCONT_3:5;

theorem Th63: :: BORSUK_5:63
for A being Subset of R^1
for a being real number st A = ].a,+infty.[ holds
A is open
proof end;

theorem Th64: :: BORSUK_5:64
for A being Subset of R^1
for a being real number st A = ].-infty,a.[ holds
A is open
proof end;

theorem Th65: :: BORSUK_5:65
for A being Subset of R^1
for a being real number st A = ].-infty,a.] holds
A is closed
proof end;

theorem Th66: :: BORSUK_5:66
for A being Subset of R^1
for a being real number st A = [.a,+infty.[ holds
A is closed
proof end;

theorem Th67: :: BORSUK_5:67
for a being real number holds [.a,+infty.[ = {a} \/ ].a,+infty.[
proof end;

theorem Th68: :: BORSUK_5:68
for a being real number holds ].-infty,a.] = {a} \/ ].-infty,a.[
proof end;

Lm15: for a being real number holds ].a,+infty.[ c= [.a,+infty.[
by LIMFUNC1:10;

Lm16: for a being real number holds ].-infty,a.[ c= ].-infty,a.]
by LIMFUNC1:15;

registration
let a be real number ;
cluster ].a,+infty.[ -> non empty ;
coherence