set Y = 1TopSp {1};
let x0, y0 be Point of (Tunit_circle 2); for xt being Point of R^1
for f being Path of x0,y0 st xt in CircleMap " {x0} holds
ex ft being Function of I[01] ,R^1 st
( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
let xt be Point of R^1 ; for f being Path of x0,y0 st xt in CircleMap " {x0} holds
ex ft being Function of I[01] ,R^1 st
( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
let f be Path of x0,y0; ( xt in CircleMap " {x0} implies ex ft being Function of I[01] ,R^1 st
( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) ) )
deffunc H1( set , Element of the carrier of I[01] ) -> Element of the carrier of (Tunit_circle 2) = f . $2;
consider F being Function of [:the carrier of (1TopSp {1}),the carrier of I[01] :],the carrier of (Tunit_circle 2) such that
A1:
for y being Element of (1TopSp {1})
for i being Element of the carrier of I[01] holds F . y,i = H1(y,i)
from BINOP_1:sch 4();
reconsider j = 1 as Point of (1TopSp {1}) by TARSKI:def 1;
A2:
[j,j0] in [:the carrier of (1TopSp {1}),{0 }:]
by Lm4, ZFMISC_1:106;
A3:
the carrier of [:(1TopSp {1}),I[01] :] = [:the carrier of (1TopSp {1}),the carrier of I[01] :]
by BORSUK_1:def 5;
then reconsider F = F as Function of [:(1TopSp {1}),I[01] :],(Tunit_circle 2) ;
set Ft = [:(1TopSp {1}),(Sspace 0[01] ):] --> xt;
A4:
the carrier of [:(1TopSp {1}),(Sspace 0[01] ):] = [:the carrier of (1TopSp {1}),the carrier of (Sspace 0[01] ):]
by BORSUK_1:def 5;
then A5:
for y being Element of (1TopSp {1})
for i being Element of {0 } holds ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . [y,i] = xt
by Lm14, FUNCOP_1:13;
A6:
[#] (1TopSp {1}) = the carrier of (1TopSp {1})
;
for p being Point of [:(1TopSp {1}),I[01] :]
for V being Subset of (Tunit_circle 2) st F . p in V & V is open holds
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & F .: W c= V )
proof
let p be
Point of
[:(1TopSp {1}),I[01] :];
for V being Subset of (Tunit_circle 2) st F . p in V & V is open holds
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & F .: W c= V )let V be
Subset of
(Tunit_circle 2);
( F . p in V & V is open implies ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & F .: W c= V ) )
assume A7:
(
F . p in V &
V is
open )
;
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & F .: W c= V )
consider p1 being
Point of
(1TopSp {1}),
p2 being
Point of
I[01] such that A8:
p = [p1,p2]
by BORSUK_1:50;
F . p1,
p2 = f . p2
by A1;
then consider S being
Subset of
I[01] such that A9:
p2 in S
and A10:
S is
open
and A11:
f .: S c= V
by A7, A8, JGRAPH_2:20;
set W =
[:{1},S:];
[:{1},S:] c= [:the carrier of (1TopSp {1}),the carrier of I[01] :]
by ZFMISC_1:118;
then reconsider W =
[:{1},S:] as
Subset of
[:(1TopSp {1}),I[01] :] by BORSUK_1:def 5;
take
W
;
( p in W & W is open & F .: W c= V )
thus
p in W
by A8, A9, ZFMISC_1:106;
( W is open & F .: W c= V )
thus
W is
open
by A6, A10, BORSUK_1:46;
F .: W c= V
let y be
set ;
TARSKI:def 3 ( not y in F .: W or y in V )
assume
y in F .: W
;
y in V
then consider x being
set such that A12:
x in the
carrier of
[:(1TopSp {1}),I[01] :]
and A13:
x in W
and A14:
y = F . x
by FUNCT_2:115;
consider x1 being
Point of
(1TopSp {1}),
x2 being
Point of
I[01] such that A15:
x = [x1,x2]
by A12, BORSUK_1:50;
x2 in S
by A13, A15, ZFMISC_1:106;
then A16:
f . x2 in f .: S
by FUNCT_2:43;
y =
F . x1,
x2
by A14, A15
.=
f . x2
by A1
;
hence
y in V
by A11, A16;
verum
end;
then A17:
F is continuous
by JGRAPH_2:20;
assume
xt in CircleMap " {x0}
; ex ft being Function of I[01] ,R^1 st
( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
then A18:
CircleMap . xt in {x0}
by FUNCT_2:46;
A19:
for x being set st x in dom (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) holds
(F | [:the carrier of (1TopSp {1}),{0 }:]) . x = (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) . x
proof
let x be
set ;
( x in dom (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) implies (F | [:the carrier of (1TopSp {1}),{0 }:]) . x = (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) . x )
assume A20:
x in dom (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt))
;
(F | [:the carrier of (1TopSp {1}),{0 }:]) . x = (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) . x
consider x1,
x2 being
set such that A21:
x1 in the
carrier of
(1TopSp {1})
and A22:
x2 in {0 }
and A23:
x = [x1,x2]
by A4, A20, Lm14, ZFMISC_1:def 2;
A24:
x2 = 0
by A22, TARSKI:def 1;
thus (F | [:the carrier of (1TopSp {1}),{0 }:]) . x =
F . x1,
x2
by A4, A20, A23, Lm14, FUNCT_1:72
.=
f . x2
by A1, A21, A24, Lm2
.=
x0
by A24, BORSUK_2:def 4
.=
CircleMap . xt
by A18, TARSKI:def 1
.=
CircleMap . (([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . x)
by A5, A21, A22, A23
.=
(CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) . x
by A20, FUNCT_1:22
;
verum
end;
A25:
dom (CircleMap * ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt)) = [:the carrier of (1TopSp {1}),{0 }:]
by A4, Lm14, FUNCT_2:def 1;
A26:
dom F = the carrier of [:(1TopSp {1}),I[01] :]
by FUNCT_2:def 1;
then A27:
[:the carrier of (1TopSp {1}),{0 }:] c= dom F
by A3, Lm3, ZFMISC_1:118;
then
dom (F | [:the carrier of (1TopSp {1}),{0 }:]) = [:the carrier of (1TopSp {1}),{0 }:]
by RELAT_1:91;
then consider G being Function of [:(1TopSp {1}),I[01] :],R^1 such that
A28:
G is continuous
and
A29:
F = CircleMap * G
and
A30:
G | [:the carrier of (1TopSp {1}),{0 }:] = [:(1TopSp {1}),(Sspace 0[01] ):] --> xt
and
A31:
for H being Function of [:(1TopSp {1}),I[01] :],R^1 st H is continuous & F = CircleMap * H & H | [:the carrier of (1TopSp {1}),{0 }:] = [:(1TopSp {1}),(Sspace 0[01] ):] --> xt holds
G = H
by A17, A25, A19, Th22, FUNCT_1:9;
take ft = Prj2 j,G; ( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
thus ft . 0 =
G . j,j0
by Def3
.=
([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . [j,j0]
by A30, A2, FUNCT_1:72
.=
xt
by A5, Lm4
; ( f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
for i being Point of I[01] holds f . i = (CircleMap * ft) . i
hence
f = CircleMap * ft
by FUNCT_2:113; ( ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )
thus
ft is continuous
by A28; for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1
let f1 be Function of I[01] ,R^1 ; ( f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt implies ft = f1 )
deffunc H2( set , Element of the carrier of I[01] ) -> Element of the carrier of R^1 = f1 . $2;
consider H being Function of [:the carrier of (1TopSp {1}),the carrier of I[01] :],the carrier of R^1 such that
A33:
for y being Element of (1TopSp {1})
for i being Element of the carrier of I[01] holds H . y,i = H2(y,i)
from BINOP_1:sch 4();
reconsider H = H as Function of [:(1TopSp {1}),I[01] :],R^1 by A3;
assume that
A34:
f1 is continuous
and
A35:
f = CircleMap * f1
and
A36:
f1 . 0 = xt
; ft = f1
for p being Point of [:(1TopSp {1}),I[01] :]
for V being Subset of R^1 st H . p in V & V is open holds
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & H .: W c= V )
proof
let p be
Point of
[:(1TopSp {1}),I[01] :];
for V being Subset of R^1 st H . p in V & V is open holds
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & H .: W c= V )let V be
Subset of
R^1 ;
( H . p in V & V is open implies ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & H .: W c= V ) )
assume A37:
(
H . p in V &
V is
open )
;
ex W being Subset of [:(1TopSp {1}),I[01] :] st
( p in W & W is open & H .: W c= V )
consider p1 being
Point of
(1TopSp {1}),
p2 being
Point of
I[01] such that A38:
p = [p1,p2]
by BORSUK_1:50;
H . p =
H . p1,
p2
by A38
.=
f1 . p2
by A33
;
then consider W being
Subset of
I[01] such that A39:
p2 in W
and A40:
W is
open
and A41:
f1 .: W c= V
by A34, A37, JGRAPH_2:20;
take W1 =
[:([#] (1TopSp {1})),W:];
( p in W1 & W1 is open & H .: W1 c= V )
thus
p in W1
by A38, A39, ZFMISC_1:106;
( W1 is open & H .: W1 c= V )
thus
W1 is
open
by A40, BORSUK_1:46;
H .: W1 c= V
let y be
set ;
TARSKI:def 3 ( not y in H .: W1 or y in V )
assume
y in H .: W1
;
y in V
then consider c being
Element of
[:(1TopSp {1}),I[01] :] such that A42:
c in W1
and A43:
y = H . c
by FUNCT_2:116;
consider c1,
c2 being
set such that A44:
c1 in [#] (1TopSp {1})
and A45:
c2 in W
and A46:
c = [c1,c2]
by A42, ZFMISC_1:def 2;
A47:
f1 . c2 in f1 .: W
by A45, FUNCT_2:43;
H . c =
H . c1,
c2
by A46
.=
f1 . c2
by A33, A44, A45
;
hence
y in V
by A41, A43, A47;
verum
end;
then A48:
H is continuous
by JGRAPH_2:20;
for x being Point of [:(1TopSp {1}),I[01] :] holds F . x = (CircleMap * H) . x
then A50:
F = CircleMap * H
by FUNCT_2:113;
for i being Point of I[01] holds ft . i = f1 . i
proof
let i be
Point of
I[01] ;
ft . i = f1 . i
A51:
dom H = the
carrier of
[:(1TopSp {1}),I[01] :]
by FUNCT_2:def 1;
then A52:
dom (H | [:the carrier of (1TopSp {1}),{0 }:]) = [:the carrier of (1TopSp {1}),{0 }:]
by A26, A27, RELAT_1:91;
A53:
for
x being
set st
x in dom (H | [:the carrier of (1TopSp {1}),{0 }:]) holds
(H | [:the carrier of (1TopSp {1}),{0 }:]) . x = ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . x
proof
let x be
set ;
( x in dom (H | [:the carrier of (1TopSp {1}),{0 }:]) implies (H | [:the carrier of (1TopSp {1}),{0 }:]) . x = ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . x )
assume A54:
x in dom (H | [:the carrier of (1TopSp {1}),{0 }:])
;
(H | [:the carrier of (1TopSp {1}),{0 }:]) . x = ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . x
then consider x1,
x2 being
set such that A55:
x1 in the
carrier of
(1TopSp {1})
and A56:
x2 in {0 }
and A57:
x = [x1,x2]
by A52, ZFMISC_1:def 2;
A58:
x2 = j0
by A56, TARSKI:def 1;
thus (H | [:the carrier of (1TopSp {1}),{0 }:]) . x =
H . x1,
x2
by A54, A57, FUNCT_1:70
.=
f1 . x2
by A33, A55, A58
.=
([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) . x
by A5, A36, A55, A56, A57, A58
;
verum
end;
dom ([:(1TopSp {1}),(Sspace 0[01] ):] --> xt) = [:the carrier of (1TopSp {1}),{0 }:]
by A4, Lm14, FUNCT_2:def 1;
then A59:
H | [:the carrier of (1TopSp {1}),{0 }:] = [:(1TopSp {1}),(Sspace 0[01] ):] --> xt
by A26, A27, A51, A53, FUNCT_1:9, RELAT_1:91;
thus ft . i =
G . j,
i
by Def3
.=
H . j,
i
by A31, A50, A48, A59
.=
f1 . i
by A33
;
verum
end;
hence
ft = f1
by FUNCT_2:113; verum