:: The Brouwer Fixed Point Theorem for Intervals
:: by Toshihiko Watanabe
::
:: Received August 17, 1992
:: Copyright (c) 1992 Association of Mizar Users

begin

Lm1: for a, b being real number
for x being set st x in [.a,b.] holds
x is Real
;

Lm2: for a, b being real number
for x being Point of (Closed-Interval-TSpace a,b) st a <= b holds
x is Real

proof end;

theorem :: TREAL_1:1

canceled;

theorem :: TREAL_1:2

canceled;

theorem :: TREAL_1:3

canceled;

theorem Th4: :: TREAL_1:4

for a, b being real number
for A being Subset of R^1 st A = [.a,b.] holds
A is closed

proof end;

theorem Th5: :: TREAL_1:5

for a, b being real number st a <= b holds
Closed-Interval-TSpace a,b is closed SubSpace of R^1

proof end;

theorem :: TREAL_1:6

for a, c, d, b being real number st a <= c & d <= b & c <= d holds
Closed-Interval-TSpace c,d is closed SubSpace of Closed-Interval-TSpace a,b

proof end;

theorem :: TREAL_1:7

for a, c, b, d being real number st a <= c & b <= d & c <= b holds
( Closed-Interval-TSpace a,d = (Closed-Interval-TSpace a,b) union (Closed-Interval-TSpace c,d) & Closed-Interval-TSpace c,b = (Closed-Interval-TSpace a,b) meet (Closed-Interval-TSpace c,d) )

proof end;

definition

let a, b be real number ;
assume A1: a <= b ;
func (#) a,b -> Point of (Closed-Interval-TSpace a,b) equals :Def1: :: TREAL_1:def 1
a;
coherence
a is Point of (Closed-Interval-TSpace a,b)

proof end;

correctness
;
func a,b (#) -> Point of (Closed-Interval-TSpace a,b) equals :Def2: :: TREAL_1:def 2
b;
coherence
b is Point of (Closed-Interval-TSpace a,b)

proof end;

correctness
;

end;

:: deftheorem Def1 defines (#) TREAL_1:def 1 :

for a, b being real number st a <= b holds
(#) a,b = a;

:: deftheorem Def2 defines (#) TREAL_1:def 2 :

for a, b being real number st a <= b holds
a,b (#) = b;

theorem :: TREAL_1:8

( 0[01] = (#) 0 ,1 & 1[01] = 0 ,1 (#) ) by Def1, Def2, BORSUK_1:def 17, BORSUK_1:def 18;

theorem :: TREAL_1:9

for a, b, c being real number st a <= b & b <= c holds
( (#) a,b = (#) a,c & b,c (#) = a,c (#) )

proof end;

begin

definition

let a, b be real number ;
assume A1: a <= b ;
let t1, t2 be Point of (Closed-Interval-TSpace a,b);
func L[01] t1,t2 -> Function of (Closed-Interval-TSpace 0 ,1),(Closed-Interval-TSpace a,b) means :Def3: :: TREAL_1:def 3
for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
it . s = ((1 - r) * r1) + (r * r2);
existence
ex b₁ being Function of (Closed-Interval-TSpace 0 ,1),(Closed-Interval-TSpace a,b) st
for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₁ . s = ((1 - r) * r1) + (r * r2)

proof end;

uniqueness
for b₁, b₂ being Function of (Closed-Interval-TSpace 0 ,1),(Closed-Interval-TSpace a,b) st ( for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₁ . s = ((1 - r) * r1) + (r * r2) ) & ( for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₂ . s = ((1 - r) * r1) + (r * r2) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines L[01] TREAL_1:def 3 :

for a, b being real number st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace a,b)
for b₅ being Function of (Closed-Interval-TSpace 0 ,1),(Closed-Interval-TSpace a,b) holds
( b₅ = L[01] t1,t2 iff for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₅ . s = ((1 - r) * r1) + (r * r2) );

theorem Th10: :: TREAL_1:10

for a, b being real number st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace a,b)
for s being Point of (Closed-Interval-TSpace 0 ,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
(L[01] t1,t2) . s = ((r2 - r1) * r) + r1

proof end;

theorem Th11: :: TREAL_1:11

for a, b being real number st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace a,b) holds L[01] t1,t2 is continuous Function of (Closed-Interval-TSpace 0 ,1),(Closed-Interval-TSpace a,b)

proof end;

theorem :: TREAL_1:12

for a, b being real number st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace a,b) holds
( (L[01] t1,t2) . ((#) 0 ,1) = t1 & (L[01] t1,t2) . (0 ,1 (#) ) = t2 )

proof end;

theorem :: TREAL_1:13

L[01] ((#) 0 ,1),(0 ,1 (#) ) = id (Closed-Interval-TSpace 0 ,1)

proof end;

definition

let a, b be real number ;
assume A1: a < b ;
let t1, t2 be Point of (Closed-Interval-TSpace 0 ,1);
func P[01] a,b,t1,t2 -> Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0 ,1) means :Def4: :: TREAL_1:def 4
for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
it . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a);
existence
ex b₁ being Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0 ,1) st
for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₁ . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a)

proof end;

uniqueness
for b₁, b₂ being Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0 ,1) st ( for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₁ . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) ) & ( for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₂ . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines P[01] TREAL_1:def 4 :

for a, b being real number st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace 0 ,1)
for b₅ being Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0 ,1) holds
( b₅ = P[01] a,b,t1,t2 iff for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b₅ . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) );

theorem Th14: :: TREAL_1:14

for a, b being real number st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace 0 ,1)
for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being Real st s = r & r1 = t1 & r2 = t2 holds
(P[01] a,b,t1,t2) . s = (((r2 - r1) / (b - a)) * r) + (((b * r1) - (a * r2)) / (b - a))

proof end;

theorem Th15: :: TREAL_1:15

for a, b being real number st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace 0 ,1) holds P[01] a,b,t1,t2 is continuous Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0 ,1)

proof end;

theorem :: TREAL_1:16

for a, b being real number st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace 0 ,1) holds
( (P[01] a,b,t1,t2) . ((#) a,b) = t1 & (P[01] a,b,t1,t2) . (a,b (#) ) = t2 )

proof end;

theorem :: TREAL_1:17

P[01] 0 ,1,((#) 0 ,1),(0 ,1 (#) ) = id (Closed-Interval-TSpace 0 ,1)

proof end;

theorem Th18: :: TREAL_1:18

for a, b being real number st a < b holds
( id (Closed-Interval-TSpace a,b) = (L[01] ((#) a,b),(a,b (#) )) * (P[01] a,b,((#) 0 ,1),(0 ,1 (#) )) & id (Closed-Interval-TSpace 0 ,1) = (P[01] a,b,((#) 0 ,1),(0 ,1 (#) )) * (L[01] ((#) a,b),(a,b (#) )) )

proof end;

theorem Th19: :: TREAL_1:19

for a, b being real number st a < b holds
( id (Closed-Interval-TSpace a,b) = (L[01] (a,b (#) ),((#) a,b)) * (P[01] a,b,(0 ,1 (#) ),((#) 0 ,1)) & id (Closed-Interval-TSpace 0 ,1) = (P[01] a,b,(0 ,1 (#) ),((#) 0 ,1)) * (L[01] (a,b (#) ),((#) a,b)) )

proof end;

theorem Th20: :: TREAL_1:20

for a, b being real number st a < b holds
( L[01] ((#) a,b),(a,b (#) ) is being_homeomorphism & (L[01] ((#) a,b),(a,b (#) )) " = P[01] a,b,((#) 0 ,1),(0 ,1 (#) ) & P[01] a,b,((#) 0 ,1),(0 ,1 (#) ) is being_homeomorphism & (P[01] a,b,((#) 0 ,1),(0 ,1 (#) )) " = L[01] ((#) a,b),(a,b (#) ) )

proof end;

theorem :: TREAL_1:21

for a, b being real number st a < b holds
( L[01] (a,b (#) ),((#) a,b) is being_homeomorphism & (L[01] (a,b (#) ),((#) a,b)) " = P[01] a,b,(0 ,1 (#) ),((#) 0 ,1) & P[01] a,b,(0 ,1 (#) ),((#) 0 ,1) is being_homeomorphism & (P[01] a,b,(0 ,1 (#) ),((#) 0 ,1)) " = L[01] (a,b (#) ),((#) a,b) )

proof end;

begin

theorem Th22: :: TREAL_1:22

I[01] is connected

proof end;

theorem :: TREAL_1:23

for a, b being real number st a <= b holds
Closed-Interval-TSpace a,b is connected

proof end;

theorem Th24: :: TREAL_1:24

for f being continuous Function of I[01] ,I[01] ex x being Point of I[01] st f . x = x

proof end;

theorem Th25: :: TREAL_1:25

for a, b being real number st a <= b holds
for f being continuous Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace a,b) ex x being Point of (Closed-Interval-TSpace a,b) st f . x = x

proof end;

theorem Th26: :: TREAL_1:26

for X, Y being non empty SubSpace of R^1
for f being continuous Function of X,Y st ex a, b being Real st
( a <= b & [.a,b.] c= the carrier of X & [.a,b.] c= the carrier of Y & f .: [.a,b.] c= [.a,b.] ) holds
ex x being Point of X st f . x = x

proof end;

theorem :: TREAL_1:27

for X, Y being non empty SubSpace of R^1
for f being continuous Function of X,Y st ex a, b being Real st
( a <= b & [.a,b.] c= the carrier of X & f .: [.a,b.] c= [.a,b.] ) holds
ex x being Point of X st f . x = x

proof end;