:: BORSUK_5 semantic presentation
theorem :: BORSUK_5:1
canceled;
theorem Th2: :: BORSUK_5:2
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}
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
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
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 )
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
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
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
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
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
Lm3:
[#] R^1 = REAL
by TOPMETR:24;
theorem :: BORSUK_5:11
canceled;
theorem :: BORSUK_5:12
canceled;
theorem :: BORSUK_5:13
canceled;
theorem Th14: :: BORSUK_5:14
theorem Th15: :: BORSUK_5:15
theorem Th16: :: BORSUK_5:16
theorem Th17: :: BORSUK_5:17
theorem Th18: :: BORSUK_5:18
theorem Th19: :: BORSUK_5:19
theorem Th20: :: BORSUK_5:20
theorem :: BORSUK_5:21
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
theorem Th27: :: BORSUK_5:27
theorem Th28: :: BORSUK_5:28
theorem Th29: :: BORSUK_5:29
theorem :: BORSUK_5:30
canceled;
theorem :: BORSUK_5:31
canceled;
theorem :: BORSUK_5:32
canceled;
theorem :: BORSUK_5:33
theorem Th34: :: BORSUK_5:34
theorem Th35: :: BORSUK_5:35
theorem Th36: :: BORSUK_5:36
:: deftheorem defines IRRAT BORSUK_5:def 3 :
:: deftheorem defines RAT BORSUK_5:def 4 :
:: deftheorem defines IRRAT BORSUK_5:def 5 :
theorem Th37: :: BORSUK_5:37
theorem Th38: :: BORSUK_5:38
theorem Th39: :: BORSUK_5:39
theorem Th40: :: BORSUK_5:40
theorem Th41: :: BORSUK_5:41
theorem Th42: :: BORSUK_5:42
theorem Th43: :: BORSUK_5:43
theorem Th44: :: BORSUK_5:44
theorem Th45: :: BORSUK_5:45
theorem :: BORSUK_5:46
canceled;
theorem Th47: :: BORSUK_5:47
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
theorem Th51: :: BORSUK_5:51
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 )
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 )
theorem Th52: :: BORSUK_5:52
theorem Th53: :: BORSUK_5:53
theorem Th54: :: BORSUK_5:54
theorem Th55: :: BORSUK_5:55
theorem Th56: :: BORSUK_5:56
theorem Th57: :: BORSUK_5:57
theorem Th58: :: BORSUK_5:58
theorem Th59: :: BORSUK_5:59
theorem :: BORSUK_5:60
theorem Th61: :: BORSUK_5:61
theorem Th62: :: BORSUK_5:62
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
theorem Th64: :: BORSUK_5:64
theorem Th65: :: BORSUK_5:65
theorem Th66: :: BORSUK_5:66
theorem Th67: :: BORSUK_5:67
theorem Th68: :: BORSUK_5:68
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;
theorem :: BORSUK_5:69
canceled;
theorem :: BORSUK_5:70
canceled;
theorem Th71: :: BORSUK_5:71
theorem :: BORSUK_5:72
theorem :: BORSUK_5:73
theorem Th74: :: BORSUK_5:74
theorem Th75: :: BORSUK_5:75
theorem :: BORSUK_5:76
theorem Th77: :: BORSUK_5:77
theorem :: BORSUK_5:78
theorem Th79: :: BORSUK_5:79
theorem :: BORSUK_5:80
theorem Th81: :: BORSUK_5:81
theorem :: BORSUK_5:82
theorem :: BORSUK_5:83
theorem Th84: :: BORSUK_5:84
Lm17:
for a, b being real number st b <= a holds
RAT a,b = {}
Lm18:
for a, b being real number st b <= a holds
REAL = ].-infty,a.] \/ [.b,+infty.[
theorem Th85: :: BORSUK_5:85
theorem Th86: :: BORSUK_5:86
theorem Th87: :: BORSUK_5:87
theorem Th88: :: BORSUK_5:88
theorem :: BORSUK_5:89
theorem :: BORSUK_5:90
Lm19:
((IRRAT 2,4) \/ {4}) \/ {5} c= ].1,+infty.[
].1,+infty.[ c= [.1,+infty.[
by Lm15;
then Lm20:
((IRRAT 2,4) \/ {4}) \/ {5} c= [.1,+infty.[
by Lm19, XBOOLE_1:1;
Lm21:
].-infty,1.[ /\ (((].-infty,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}) = ].-infty,1.[
theorem Th91: :: BORSUK_5:91
Lm22:
].1,+infty.[ /\ (((].-infty,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}) = ((].1,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}
theorem :: BORSUK_5:92
theorem Th93: :: BORSUK_5:93
theorem Th94: :: BORSUK_5:94
theorem :: BORSUK_5:95
theorem :: BORSUK_5:96
theorem Th97: :: BORSUK_5:97
theorem :: BORSUK_5:98
theorem :: BORSUK_5:99
theorem Th100: :: BORSUK_5:100
theorem :: BORSUK_5:101
theorem :: BORSUK_5:102
theorem :: BORSUK_5:103
theorem :: BORSUK_5:104
theorem :: BORSUK_5:105
theorem :: BORSUK_5:106
theorem Th107: :: BORSUK_5:107
theorem Th108: :: BORSUK_5:108
theorem Th109: :: BORSUK_5:109
theorem Th110: :: BORSUK_5:110
theorem Th111: :: BORSUK_5:111
theorem Th112: :: BORSUK_5:112
theorem :: BORSUK_5:113
:: deftheorem Def6 defines with_proper_subsets BORSUK_5:def 6 :
theorem :: BORSUK_5:114
theorem :: BORSUK_5:115