let a, b, c, d be Real; for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[a,c]|,|[a,d]|) holds
( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 )
let p1, p2 be Point of (TOP-REAL 2); ( a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[a,c]|,|[a,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 ) )
set K = rectangle (a,b,c,d);
assume that
A1:
a < b
and
A2:
c < d
and
A3:
p1 in LSeg (|[a,c]|,|[a,d]|)
and
A4:
p2 in LSeg (|[a,c]|,|[a,d]|)
; ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 )
A5:
rectangle (a,b,c,d) is being_simple_closed_curve
by A1, A2, Th50;
A6:
p1 `1 = a
by A2, A3, Th1;
A7:
c <= p1 `2
by A2, A3, Th1;
A8:
p2 `1 = a
by A2, A4, Th1;
A9:
E-max (rectangle (a,b,c,d)) = |[b,d]|
by A1, A2, Th46;
A10:
Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))
by A1, A2, Th51;
then A11:
LSeg (|[a,c]|,|[a,d]|) c= Upper_Arc (rectangle (a,b,c,d))
by XBOOLE_1:7;
A12:
(Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))}
by A5, JORDAN6:def 9;
A13:
now ( p2 in Lower_Arc (rectangle (a,b,c,d)) implies p2 = W-min (rectangle (a,b,c,d)) )assume
p2 in Lower_Arc (rectangle (a,b,c,d))
;
p2 = W-min (rectangle (a,b,c,d))then A14:
p2 in (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d)))
by A4, A11, XBOOLE_0:def 4;
hence
p2 = W-min (rectangle (a,b,c,d))
by A12, A14, TARSKI:def 2;
verum end;
thus
( LE p1,p2, rectangle (a,b,c,d) implies p1 `2 <= p2 `2 )
( p1 `2 <= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) )proof
assume
LE p1,
p2,
rectangle (
a,
b,
c,
d)
;
p1 `2 <= p2 `2
then A15:
( (
p1 in Upper_Arc (rectangle (a,b,c,d)) &
p2 in Lower_Arc (rectangle (a,b,c,d)) & not
p2 = W-min (rectangle (a,b,c,d)) ) or (
p1 in Upper_Arc (rectangle (a,b,c,d)) &
p2 in Upper_Arc (rectangle (a,b,c,d)) &
LE p1,
p2,
Upper_Arc (rectangle (a,b,c,d)),
W-min (rectangle (a,b,c,d)),
E-max (rectangle (a,b,c,d)) ) or (
p1 in Lower_Arc (rectangle (a,b,c,d)) &
p2 in Lower_Arc (rectangle (a,b,c,d)) & not
p2 = W-min (rectangle (a,b,c,d)) &
LE p1,
p2,
Lower_Arc (rectangle (a,b,c,d)),
E-max (rectangle (a,b,c,d)),
W-min (rectangle (a,b,c,d)) ) )
by JORDAN6:def 10;
consider f being
Function of
I[01],
((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) such that A16:
f is
being_homeomorphism
and A17:
f . 0 = W-min (rectangle (a,b,c,d))
and A18:
f . 1
= E-max (rectangle (a,b,c,d))
and
rng f = Upper_Arc (rectangle (a,b,c,d))
and
for
r being
Real st
r in [.0,(1 / 2).] holds
f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|)
and
for
r being
Real st
r in [.(1 / 2),1.] holds
f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|)
and A19:
for
p being
Point of
(TOP-REAL 2) st
p in LSeg (
|[a,c]|,
|[a,d]|) holds
(
0 <= (((p `2) - c) / (d - c)) / 2 &
(((p `2) - c) / (d - c)) / 2
<= 1 &
f . ((((p `2) - c) / (d - c)) / 2) = p )
and
for
p being
Point of
(TOP-REAL 2) st
p in LSeg (
|[a,d]|,
|[b,d]|) holds
(
0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) &
((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 &
f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p )
by A1, A2, Th53;
reconsider s1 =
(((p1 `2) - c) / (d - c)) / 2,
s2 =
(((p2 `2) - c) / (d - c)) / 2 as
Real ;
A20:
f . s1 = p1
by A3, A19;
A21:
f . s2 = p2
by A4, A19;
A22:
d - c > 0
by A2, XREAL_1:50;
A23:
s1 <= 1
by A3, A19;
A24:
0 <= s2
by A4, A19;
s2 <= 1
by A4, A19;
then
s1 <= s2
by A13, A15, A16, A17, A18, A20, A21, A23, A24, JORDAN5C:def 3;
then
((((p1 `2) - c) / (d - c)) / 2) * 2
<= ((((p2 `2) - c) / (d - c)) / 2) * 2
by XREAL_1:64;
then
(((p1 `2) - c) / (d - c)) * (d - c) <= (((p2 `2) - c) / (d - c)) * (d - c)
by A22, XREAL_1:64;
then
(p1 `2) - c <= (((p2 `2) - c) / (d - c)) * (d - c)
by A22, XCMPLX_1:87;
then
(p1 `2) - c <= (p2 `2) - c
by A22, XCMPLX_1:87;
then
((p1 `2) - c) + c <= ((p2 `2) - c) + c
by XREAL_1:7;
hence
p1 `2 <= p2 `2
;
verum
end;
thus
( p1 `2 <= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) )
verumproof
assume A25:
p1 `2 <= p2 `2
;
LE p1,p2, rectangle (a,b,c,d)
for
g being
Function of
I[01],
((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) for
s1,
s2 being
Real st
g is
being_homeomorphism &
g . 0 = W-min (rectangle (a,b,c,d)) &
g . 1
= E-max (rectangle (a,b,c,d)) &
g . s1 = p1 &
0 <= s1 &
s1 <= 1 &
g . s2 = p2 &
0 <= s2 &
s2 <= 1 holds
s1 <= s2
proof
let g be
Function of
I[01],
((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d))));
for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2let s1,
s2 be
Real;
( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that A26:
g is
being_homeomorphism
and A27:
g . 0 = W-min (rectangle (a,b,c,d))
and
g . 1
= E-max (rectangle (a,b,c,d))
and A28:
g . s1 = p1
and A29:
0 <= s1
and A30:
s1 <= 1
and A31:
g . s2 = p2
and A32:
0 <= s2
and A33:
s2 <= 1
;
s1 <= s2
A34:
dom g = the
carrier of
I[01]
by FUNCT_2:def 1;
A35:
g is
one-to-one
by A26, TOPS_2:def 5;
A36:
the
carrier of
((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d))
by PRE_TOPC:8;
then reconsider g1 =
g as
Function of
I[01],
(TOP-REAL 2) by FUNCT_2:7;
g is
continuous
by A26, TOPS_2:def 5;
then A37:
g1 is
continuous
by PRE_TOPC:26;
reconsider h1 =
proj1 as
Function of
(TOP-REAL 2),
R^1 by TOPMETR:17;
reconsider h2 =
proj2 as
Function of
(TOP-REAL 2),
R^1 by TOPMETR:17;
reconsider hh1 =
h1 as
Function of
TopStruct(# the
carrier of
(TOP-REAL 2), the
topology of
(TOP-REAL 2) #),
R^1 ;
reconsider hh2 =
h2 as
Function of
TopStruct(# the
carrier of
(TOP-REAL 2), the
topology of
(TOP-REAL 2) #),
R^1 ;
A38:
TopStruct(# the
carrier of
(TOP-REAL 2), the
topology of
(TOP-REAL 2) #) =
TopStruct(# the
carrier of
(TOP-REAL 2), the
topology of
(TOP-REAL 2) #)
| ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #))
by TSEP_1:3
.=
TopStruct(# the
carrier of
((TOP-REAL 2) | ([#] (TOP-REAL 2))), the
topology of
((TOP-REAL 2) | ([#] (TOP-REAL 2))) #)
by PRE_TOPC:36
.=
(TOP-REAL 2) | ([#] (TOP-REAL 2))
;
then
( ( for
p being
Point of
((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds
hh1 . p = proj1 . p ) implies
hh1 is
continuous )
by JGRAPH_2:29;
then A39:
( ( for
p being
Point of
((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds
hh1 . p = proj1 . p ) implies
h1 is
continuous )
by PRE_TOPC:32;
( ( for
p being
Point of
((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds
hh2 . p = proj2 . p ) implies
hh2 is
continuous )
by A38, JGRAPH_2:30;
then
( ( for
p being
Point of
((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds
hh2 . p = proj2 . p ) implies
h2 is
continuous )
by PRE_TOPC:32;
then consider h being
Function of
(TOP-REAL 2),
R^1 such that A40:
for
p being
Point of
(TOP-REAL 2) for
r1,
r2 being
Real st
hh1 . p = r1 &
hh2 . p = r2 holds
h . p = r1 + r2
and A41:
h is
continuous
by A39, JGRAPH_2:19;
reconsider k =
h * g1 as
Function of
I[01],
R^1 ;
A42:
W-min (rectangle (a,b,c,d)) = |[a,c]|
by A1, A2, Th46;
now not s1 > s2assume A43:
s1 > s2
;
contradictionA44:
dom g = [.0,1.]
by BORSUK_1:40, FUNCT_2:def 1;
0 in [.0,1.]
by XXREAL_1:1;
then A45:
k . 0 =
h . (W-min (rectangle (a,b,c,d)))
by A27, A44, FUNCT_1:13
.=
(h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d))))
by A40
.=
((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d))))
by PSCOMP_1:def 5
.=
((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2)
by PSCOMP_1:def 6
.=
a + ((W-min (rectangle (a,b,c,d))) `2)
by A42, EUCLID:52
.=
a + c
by A42, EUCLID:52
;
s1 in [.0,1.]
by A29, A30, XXREAL_1:1;
then A46:
k . s1 =
h . p1
by A28, A44, FUNCT_1:13
.=
(h1 . p1) + (h2 . p1)
by A40
.=
(p1 `1) + (proj2 . p1)
by PSCOMP_1:def 5
.=
a + (p1 `2)
by A6, PSCOMP_1:def 6
;
A47:
s2 in [.0,1.]
by A32, A33, XXREAL_1:1;
then A48:
k . s2 =
h . p2
by A31, A44, FUNCT_1:13
.=
(h1 . p2) + (h2 . p2)
by A40
.=
(p2 `1) + (proj2 . p2)
by PSCOMP_1:def 5
.=
a + (p2 `2)
by A8, PSCOMP_1:def 6
;
A49:
k . 0 <= k . s1
by A7, A45, A46, XREAL_1:7;
A50:
k . s1 <= k . s2
by A25, A46, A48, XREAL_1:7;
A51:
0 in [.0,1.]
by XXREAL_1:1;
then A52:
[.0,s2.] c= [.0,1.]
by A47, XXREAL_2:def 12;
reconsider B =
[.0,s2.] as
Subset of
I[01] by A47, A51, BORSUK_1:40, XXREAL_2:def 12;
A53:
B is
connected
by A32, A47, A51, BORSUK_1:40, BORSUK_4:24;
A54:
0 in B
by A32, XXREAL_1:1;
A55:
s2 in B
by A32, XXREAL_1:1;
consider xc being
Point of
I[01] such that A56:
xc in B
and A57:
k . xc = k . s1
by A37, A41, A49, A50, A53, A54, A55, TOPREAL5:5;
reconsider rxc =
xc as
Real ;
A58:
for
x1,
x2 being
set st
x1 in dom k &
x2 in dom k &
k . x1 = k . x2 holds
x1 = x2
proof
let x1,
x2 be
set ;
( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 )
assume that A59:
x1 in dom k
and A60:
x2 in dom k
and A61:
k . x1 = k . x2
;
x1 = x2
reconsider r1 =
x1 as
Point of
I[01] by A59;
reconsider r2 =
x2 as
Point of
I[01] by A60;
A62:
k . x1 =
h . (g1 . x1)
by A59, FUNCT_1:12
.=
(h1 . (g1 . r1)) + (h2 . (g1 . r1))
by A40
.=
((g1 . r1) `1) + (proj2 . (g1 . r1))
by PSCOMP_1:def 5
.=
((g1 . r1) `1) + ((g1 . r1) `2)
by PSCOMP_1:def 6
;
A63:
k . x2 =
h . (g1 . x2)
by A60, FUNCT_1:12
.=
(h1 . (g1 . r2)) + (h2 . (g1 . r2))
by A40
.=
((g1 . r2) `1) + (proj2 . (g1 . r2))
by PSCOMP_1:def 5
.=
((g1 . r2) `1) + ((g1 . r2) `2)
by PSCOMP_1:def 6
;
A64:
g . r1 in Upper_Arc (rectangle (a,b,c,d))
by A36;
A65:
g . r2 in Upper_Arc (rectangle (a,b,c,d))
by A36;
reconsider gr1 =
g . r1 as
Point of
(TOP-REAL 2) by A64;
reconsider gr2 =
g . r2 as
Point of
(TOP-REAL 2) by A65;
now ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) )per cases
( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) )
by A10, A36, XBOOLE_0:def 3;
case A66:
(
g . r1 in LSeg (
|[a,c]|,
|[a,d]|) &
g . r2 in LSeg (
|[a,c]|,
|[a,d]|) )
;
x1 = x2then A67:
gr1 `1 = a
by A2, Th1;
gr2 `1 = a
by A2, A66, Th1;
then
|[(gr1 `1),(gr1 `2)]| = g . r2
by A61, A62, A63, A67, EUCLID:53;
then
g . r1 = g . r2
by EUCLID:53;
hence
x1 = x2
by A34, A35, FUNCT_1:def 4;
verum end; case A68:
(
g . r1 in LSeg (
|[a,c]|,
|[a,d]|) &
g . r2 in LSeg (
|[a,d]|,
|[b,d]|) )
;
x1 = x2then A69:
gr1 `1 = a
by A2, Th1;
A70:
gr1 `2 <= d
by A2, A68, Th1;
A71:
gr2 `2 = d
by A1, A68, Th3;
A72:
a <= gr2 `1
by A1, A68, Th3;
A73:
a + (gr1 `2) = (gr2 `1) + d
by A1, A61, A62, A63, A68, A69, Th3;
then
|[(gr1 `1),(gr1 `2)]| = g . r2
by A69, A71, A74, EUCLID:53;
then
g . r1 = g . r2
by EUCLID:53;
hence
x1 = x2
by A34, A35, FUNCT_1:def 4;
verum end; case A75:
(
g . r1 in LSeg (
|[a,d]|,
|[b,d]|) &
g . r2 in LSeg (
|[a,c]|,
|[a,d]|) )
;
x1 = x2then A76:
gr2 `1 = a
by A2, Th1;
A77:
gr2 `2 <= d
by A2, A75, Th1;
A78:
gr1 `2 = d
by A1, A75, Th3;
A79:
a <= gr1 `1
by A1, A75, Th3;
A80:
a + (gr2 `2) = (gr1 `1) + d
by A1, A61, A62, A63, A75, A76, Th3;
then
|[(gr2 `1),(gr2 `2)]| = g . r1
by A76, A78, A81, EUCLID:53;
then
g . r1 = g . r2
by EUCLID:53;
hence
x1 = x2
by A34, A35, FUNCT_1:def 4;
verum end; case A82:
(
g . r1 in LSeg (
|[a,d]|,
|[b,d]|) &
g . r2 in LSeg (
|[a,d]|,
|[b,d]|) )
;
x1 = x2then A83:
gr1 `2 = d
by A1, Th3;
gr2 `2 = d
by A1, A82, Th3;
then
|[(gr1 `1),(gr1 `2)]| = g . r2
by A61, A62, A63, A83, EUCLID:53;
then
g . r1 = g . r2
by EUCLID:53;
hence
x1 = x2
by A34, A35, FUNCT_1:def 4;
verum end; end; end;
hence
x1 = x2
;
verum
end; A84:
dom k = [.0,1.]
by BORSUK_1:40, FUNCT_2:def 1;
then
s1 in dom k
by A29, A30, XXREAL_1:1;
then
rxc = s1
by A52, A56, A57, A58, A84;
hence
contradiction
by A43, A56, XXREAL_1:1;
verum end;
hence
s1 <= s2
;
verum
end;
then
LE p1,
p2,
Upper_Arc (rectangle (a,b,c,d)),
W-min (rectangle (a,b,c,d)),
E-max (rectangle (a,b,c,d))
by A3, A4, A11, JORDAN5C:def 3;
hence
LE p1,
p2,
rectangle (
a,
b,
c,
d)
by A3, A4, A11, JORDAN6:def 10;
verum
end;