topreal2.miz
begin
reserve a for
set;
reserve p,p1,p2,q,q1,q2 for
Point of (
TOP-REAL 2);
reserve h1,h2 for
FinSequence of (
TOP-REAL 2);
Lm1: for x,X be
set st not x
in X holds (
{x}
/\ X)
=
{} by
XBOOLE_0:def 7,
ZFMISC_1: 50;
Lm2: ((
LSeg (
|[
0 ,
0 ]|,
|[1,
0 ]|))
/\ (
LSeg (
|[
0 , 1]|,
|[1, 1]|)))
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
Lm3: ((
LSeg (
|[
0 ,
0 ]|,
|[
0 , 1]|))
/\ (
LSeg (
|[1,
0 ]|,
|[1, 1]|)))
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
set p00 =
|[
0 ,
0 ]|, p01 =
|[
0 , 1]|, p10 =
|[1,
0 ]|, p11 =
|[1, 1]|, L1 = (
LSeg (p00,p01)), L2 = (
LSeg (p01,p11)), L3 = (
LSeg (p00,p10)), L4 = (
LSeg (p10,p11));
Lm4: (p00
`1 )
=
0 by
EUCLID: 52;
Lm5: (p00
`2 )
=
0 by
EUCLID: 52;
Lm6: (p01
`1 )
=
0 by
EUCLID: 52;
Lm7: (p01
`2 )
= 1 by
EUCLID: 52;
Lm8: (p10
`1 )
= 1 by
EUCLID: 52;
Lm9: (p10
`2 )
=
0 by
EUCLID: 52;
Lm10: (p11
`1 )
= 1 by
EUCLID: 52;
Lm11: (p11
`2 )
= 1 by
EUCLID: 52;
Lm12: not p00
in L4 by
Lm4,
Lm8,
Lm10,
TOPREAL1: 3;
Lm13: not p00
in L2 by
Lm5,
Lm7,
Lm11,
TOPREAL1: 4;
Lm14: not p01
in L3 by
Lm5,
Lm7,
Lm9,
TOPREAL1: 4;
Lm15: not p01
in L4 by
Lm6,
Lm8,
Lm10,
TOPREAL1: 3;
Lm16: not p10
in L1 by
Lm4,
Lm6,
Lm8,
TOPREAL1: 3;
Lm17: not p10
in L2 by
Lm7,
Lm9,
Lm11,
TOPREAL1: 4;
Lm18: not p11
in L1 by
Lm4,
Lm6,
Lm10,
TOPREAL1: 3;
Lm19: not p11
in L3 by
Lm5,
Lm9,
Lm11,
TOPREAL1: 4;
Lm20: p00
in L1 by
RLTOPSP1: 68;
Lm21: p00
in L3 by
RLTOPSP1: 68;
Lm22: p01
in L1 by
RLTOPSP1: 68;
Lm23: p01
in L2 by
RLTOPSP1: 68;
Lm24: p10
in L3 by
RLTOPSP1: 68;
Lm25: p10
in L4 by
RLTOPSP1: 68;
Lm26: p11
in L2 by
RLTOPSP1: 68;
Lm27: p11
in L4 by
RLTOPSP1: 68;
set L = { p : (p
`1 )
=
0 & (p
`2 )
<= 1 & (p
`2 )
>=
0 or (p
`1 )
<= 1 & (p
`1 )
>=
0 & (p
`2 )
= 1 or (p
`1 )
<= 1 & (p
`1 )
>=
0 & (p
`2 )
=
0 or (p
`1 )
= 1 & (p
`2 )
<= 1 & (p
`2 )
>=
0 };
Lm28: p00
in L by
Lm4,
Lm5;
Lm29: p11
in L by
Lm10,
Lm11;
Lm30: p1
<> p2 & p2
in
R^2-unit_square & p1
in (
LSeg (p00,p01)) implies ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) &
R^2-unit_square
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume that
A1: p1
<> p2 and
A2: p2
in
R^2-unit_square and
A3: p1
in (
LSeg (p00,p01));
A4: (
LSeg (p00,p1))
c= L1 by
A3,
Lm20,
TOPREAL1: 6;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then p00
in ((
LSeg (p1,p00))
/\ L3) by
Lm21,
XBOOLE_0:def 4;
then
A5:
{p00}
c= ((
LSeg (p1,p00))
/\ L3) by
ZFMISC_1: 31;
A6: ((
LSeg (p1,p00))
/\ L3)
c= (L1
/\ L3) by
A3,
Lm20,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A7: ((
LSeg (p1,p00))
/\ L3)
=
{p00} by
A5,
TOPREAL1: 17,
XBOOLE_0:def 10;
A8: (L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A9: ((
LSeg (p1,p00))
/\ L4)
=
{} by
A4,
XBOOLE_1: 3,
XBOOLE_1: 26;
p01
in (
LSeg (p01,p1)) by
RLTOPSP1: 68;
then p01
in ((
LSeg (p01,p1))
/\ L2) by
Lm23,
XBOOLE_0:def 4;
then
A10:
{p01}
c= ((
LSeg (p01,p1))
/\ L2) by
ZFMISC_1: 31;
A11: p2
in (L1
\/ L2) or p2
in (L3
\/ L4) by
A2,
TOPREAL1:def 2,
XBOOLE_0:def 3;
A12: ((
LSeg (p01,p1))
/\ L2)
c=
{p01} by
A3,
Lm22,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
A13: (
LSeg (p1,p01))
c= L1 by
A3,
Lm22,
TOPREAL1: 6;
then
A14: ((
LSeg (p01,p1))
/\ L4)
=
{} by
A8,
XBOOLE_1: 3,
XBOOLE_1: 26;
consider p such that
A15: p
= p1 and
A16: (p
`1 )
=
0 and
A17: (p
`2 )
<= 1 and
A18: (p
`2 )
>=
0 by
A3,
TOPREAL1: 13;
per cases by
A11,
XBOOLE_0:def 3;
suppose
A19: p2
in L1;
then
A20: (
LSeg (p2,p1))
c= L1 by
A3,
TOPREAL1: 6;
A21: p
=
|[(p
`1 ), (p
`2 )]| by
EUCLID: 53;
consider q such that
A22: q
= p2 and
A23: (q
`1 )
=
0 and
A24: (q
`2 )
<= 1 and
A25: (q
`2 )
>=
0 by
A19,
TOPREAL1: 13;
A26: q
=
|[(q
`1 ), (q
`2 )]| by
EUCLID: 53;
now
per cases by
A1,
A15,
A16,
A22,
A23,
A21,
A26,
XXREAL_0: 1;
case
A27: (p
`2 )
< (q
`2 );
A28: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
c=
{p1}
proof
let a be
object;
assume
A29: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A30: p
in (
LSeg (p00,p1)) by
A29,
XBOOLE_0:def 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then
A31: (p
`2 )
<= (p1
`2 ) by
A30,
TOPREAL1: 4;
A32: p
in (
LSeg (p1,p2)) by
A29,
XBOOLE_0:def 4;
then (p1
`2 )
<= (p
`2 ) by
A15,
A22,
A27,
TOPREAL1: 4;
then
A33: (p1
`2 )
= (p
`2 ) by
A31,
XXREAL_0: 1;
(p1
`1 )
<= (p
`1 ) by
A15,
A16,
A22,
A23,
A32,
TOPREAL1: 3;
then (p
`1 )
=
0 by
A15,
A16,
A22,
A23,
A32,
TOPREAL1: 3;
then p
=
|[
0 , (p1
`2 )]| by
A33,
EUCLID: 53
.= p1 by
A15,
A16,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A34: ((
LSeg (p01,p2))
/\ L2)
c= (L1
/\ L2) by
A19,
Lm22,
TOPREAL1: 6,
XBOOLE_1: 26;
A35:
now
set a = the
Element of ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)));
assume
A36: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A37: p
in (
LSeg (p00,p1)) by
A36,
XBOOLE_0:def 4;
A38: p
in (
LSeg (p2,p01)) by
A36,
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A22,
A24,
EUCLID: 52;
then
A39: (p2
`2 )
<= (p
`2 ) by
A38,
TOPREAL1: 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p
`2 )
<= (p1
`2 ) by
A37,
TOPREAL1: 4;
hence contradiction by
A15,
A22,
A27,
A39,
XXREAL_0: 2;
end;
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then p01
in ((
LSeg (p01,p2))
/\ L2) by
Lm23,
XBOOLE_0:def 4;
then
A40:
{p01}
c= ((
LSeg (p01,p2))
/\ L2) by
ZFMISC_1: 31;
now
assume p00
in ((
LSeg (p01,p2))
/\ L3);
then
A41: p00
in (
LSeg (p2,p01)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A22,
A24,
EUCLID: 52;
hence contradiction by
A18,
A22,
A27,
A41,
Lm5,
TOPREAL1: 4;
end;
then
A42:
{p00}
<> ((
LSeg (p01,p2))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ L3)
c=
{p00} by
A19,
Lm22,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then
A43: ((
LSeg (p01,p2))
/\ L3)
=
{} by
A42,
ZFMISC_1: 33;
A44: ((
LSeg (p1,p2))
/\ L3)
c=
{p00} by
A3,
A19,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
A45: ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)))
c=
{p2}
proof
let a be
object;
assume
A46: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A47: p
in (
LSeg (p2,p01)) by
A46,
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A22,
A24,
EUCLID: 52;
then
A48: (p2
`2 )
<= (p
`2 ) by
A47,
TOPREAL1: 4;
A49: p
in (
LSeg (p1,p2)) by
A46,
XBOOLE_0:def 4;
then (p
`2 )
<= (p2
`2 ) by
A15,
A22,
A27,
TOPREAL1: 4;
then
A50: (p2
`2 )
= (p
`2 ) by
A48,
XXREAL_0: 1;
(p1
`1 )
<= (p
`1 ) by
A15,
A16,
A22,
A23,
A49,
TOPREAL1: 3;
then (p
`1 )
=
0 by
A15,
A16,
A22,
A23,
A49,
TOPREAL1: 3;
then p
=
|[
0 , (p2
`2 )]| by
A50,
EUCLID: 53
.= p2 by
A22,
A23,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A51: ((
LSeg (p1,p00))
/\ L2)
c=
{p01} by
A3,
Lm20,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
now
assume p01
in ((
LSeg (p1,p00))
/\ L2);
then
A52: p01
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p01
`2 )
<= (p1
`2 ) by
A52,
TOPREAL1: 4;
hence contradiction by
A15,
A17,
A24,
A27,
Lm7,
XXREAL_0: 1;
end;
then
A53:
{p01}
<> ((
LSeg (p1,p00))
/\ L2) by
ZFMISC_1: 31;
set P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p00))
\/ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p2))));
A54: p1
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
A55: (
LSeg (p01,p2))
c= L1 by
A19,
Lm22,
TOPREAL1: 6;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00))) by
A54,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00))) by
ZFMISC_1: 31;
then
A56: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
=
{p1} by
A28,
XBOOLE_0:def 10;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A57: ((L3
\/ L4)
/\ (
LSeg (p11,p01)))
= (
{}
\/
{p11}) by
Lm2,
TOPREAL1: 18,
XBOOLE_1: 23
.=
{p11};
(L3
\/ L4)
is_an_arc_of (p00,p11) by
Lm4,
Lm8,
TOPREAL1: 12,
TOPREAL1: 16;
then
A58: ((L3
\/ L4)
\/ (
LSeg (p11,p01)))
is_an_arc_of (p00,p01) by
A57,
TOPREAL1: 10;
(((L3
\/ L4)
\/ (
LSeg (p11,p01)))
/\ (
LSeg (p01,p2)))
= (((
LSeg (p01,p2))
/\ (L3
\/ L4))
\/ ((
LSeg (p01,p2))
/\ (
LSeg (p11,p01)))) by
XBOOLE_1: 23
.= ((
{}
\/ ((
LSeg (p01,p2))
/\ L4))
\/ ((
LSeg (p01,p2))
/\ (
LSeg (p11,p01)))) by
A43,
XBOOLE_1: 23
.= (
{}
\/ ((
LSeg (p01,p2))
/\ (
LSeg (p11,p01)))) by
A55,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26
.=
{p01} by
A40,
A34,
TOPREAL1: 15,
XBOOLE_0:def 10;
then
A59: (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p2)))
is_an_arc_of (p00,p2) by
A58,
TOPREAL1: 10;
((
LSeg (p1,p00))
/\ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p00))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p00))
/\ L3)
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.=
{p00} by
A9,
A7,
A35,
A51,
A53,
ZFMISC_1: 33;
hence ((
LSeg (p1,p00))
\/ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p2))))
is_an_arc_of (p1,p2) by
A59,
TOPREAL1: 11;
(((
LSeg (p01,p2))
\/ (
LSeg (p2,p1)))
\/ (
LSeg (p1,p00)))
= L1 by
A3,
A19,
TOPREAL1: 7;
hence
R^2-unit_square
= ((((
LSeg (p1,p2))
\/ (
LSeg (p01,p2)))
\/ (
LSeg (p1,p00)))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01)))) by
TOPREAL1:def 2,
XBOOLE_1: 4
.= (((
LSeg (p1,p2))
\/ ((
LSeg (p1,p00))
\/ (
LSeg (p01,p2))))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01)))) by
XBOOLE_1: 4
.= ((
LSeg (p1,p2))
\/ (((
LSeg (p1,p00))
\/ (
LSeg (p01,p2)))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A60: p2
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))) by
A60,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
then
A61: ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)))
=
{p2} by
A45,
XBOOLE_0:def 10;
A62: (
LSeg (p1,p2))
c= L1 by
A3,
A19,
TOPREAL1: 6;
A63: (P1
/\ P2)
= (((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p1,p2))
/\ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p2))))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
\/ (((
LSeg (p1,p2))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
\/ ((((
LSeg (p1,p2))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L3)
\/ ((
LSeg (p1,p2))
/\ L4))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))))) by
A56,
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L3)
\/
{} )
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))))) by
A62,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L3)
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2}))) by
A61,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L3))
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
XBOOLE_1: 4;
A64:
now
per cases ;
suppose
A65: p1
= p00;
then p00
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L3)
<>
{} by
Lm21,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L3)
=
{p1} by
A44,
A65,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
A63;
end;
suppose
A66: p1
<> p00;
now
assume p00
in ((
LSeg (p1,p2))
/\ L3);
then p00
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p1
`2 )
<= (p00
`2 ) by
A15,
A22,
A27,
TOPREAL1: 4;
then (p00
`2 )
= (p1
`2 ) by
A3,
Lm5,
Lm7,
TOPREAL1: 4;
hence contradiction by
A15,
A16,
A66,
Lm5,
EUCLID: 53;
end;
then ((
LSeg (p1,p2))
/\ L3)
<>
{p00} by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L3)
=
{} by
A44,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
A63;
end;
end;
A67: ((
LSeg (p1,p2))
/\ L2)
c=
{p01} by
A3,
A19,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
now
per cases ;
suppose
A68: p2
<> p01;
now
assume p01
in ((
LSeg (p1,p2))
/\ L2);
then p01
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then
A69: (p01
`2 )
<= (p2
`2 ) by
A15,
A22,
A27,
TOPREAL1: 4;
(p2
`2 )
<= (p01
`2 ) by
A19,
Lm5,
Lm7,
TOPREAL1: 4;
then
A70: (p01
`2 )
= (p2
`2 ) by
A69,
XXREAL_0: 1;
p2
=
|[(p2
`1 ), (p2
`2 )]| by
EUCLID: 53
.=
|[
0 , 1]| by
A22,
A23,
A70,
EUCLID: 52;
hence contradiction by
A68;
end;
then ((
LSeg (p1,p2))
/\ L2)
<>
{p01} by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L2)
=
{} by
A67,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A64,
ENUMSET1: 1;
end;
suppose
A71: p2
= p01;
then p01
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L2)
<>
{} by
Lm23,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L2)
=
{p2} by
A67,
A71,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A64,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
case
A72: (p
`2 )
> (q
`2 );
A73: ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1)))
c=
{p1}
proof
let a be
object;
assume
A74: a
in ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A75: p
in (
LSeg (p1,p01)) by
A74,
XBOOLE_0:def 4;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
then
A76: (p1
`2 )
<= (p
`2 ) by
A75,
TOPREAL1: 4;
A77: p
in (
LSeg (p2,p1)) by
A74,
XBOOLE_0:def 4;
then (p
`2 )
<= (p1
`2 ) by
A15,
A22,
A72,
TOPREAL1: 4;
then
A78: (p1
`2 )
= (p
`2 ) by
A76,
XXREAL_0: 1;
(p2
`1 )
<= (p
`1 ) by
A15,
A16,
A22,
A23,
A77,
TOPREAL1: 3;
then (p
`1 )
=
0 by
A15,
A16,
A22,
A23,
A77,
TOPREAL1: 3;
then p
=
|[
0 , (p1
`2 )]| by
A78,
EUCLID: 53
.= p1 by
A15,
A16,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A79: (
LSeg (p2,p00))
c= L1 by
A19,
Lm20,
TOPREAL1: 6;
A80:
now
set a = the
Element of ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)));
assume
A81: ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A82: p
in (
LSeg (p00,p2)) by
A81,
XBOOLE_0:def 4;
A83: p
in (
LSeg (p1,p01)) by
A81,
XBOOLE_0:def 4;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
then
A84: (p1
`2 )
<= (p
`2 ) by
A83,
TOPREAL1: 4;
(p00
`2 )
<= (p2
`2 ) by
A22,
A25,
EUCLID: 52;
then (p
`2 )
<= (p2
`2 ) by
A82,
TOPREAL1: 4;
hence contradiction by
A15,
A22,
A72,
A84,
XXREAL_0: 2;
end;
A85: ((
LSeg (p2,p1))
/\ L3)
c=
{p00} by
A3,
A19,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
now
assume p01
in ((
LSeg (p2,p00))
/\ L2);
then
A86: p01
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p2
`2 ) by
A22,
A25,
EUCLID: 52;
then (p01
`2 )
<= (p2
`2 ) by
A86,
TOPREAL1: 4;
hence contradiction by
A17,
A22,
A24,
A72,
Lm7,
XXREAL_0: 1;
end;
then
A87:
{p01}
<> ((
LSeg (p2,p00))
/\ L2) by
ZFMISC_1: 31;
A88: ((
LSeg (p2,p00))
/\ L3)
c=
{p00} by
A19,
Lm20,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
now
assume p00
in ((
LSeg (p01,p1))
/\ L3);
then
A89: p00
in (
LSeg (p1,p01)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
hence contradiction by
A15,
A25,
A72,
A89,
Lm5,
TOPREAL1: 4;
end;
then
A90:
{p00}
<> ((
LSeg (p01,p1))
/\ L3) by
ZFMISC_1: 31;
A91: ((
LSeg (p2,p00))
/\ L2)
c=
{p01} by
A19,
Lm20,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
A92: ((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)))
c=
{p2}
proof
let a be
object;
assume
A93: a
in ((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A94: p
in (
LSeg (p00,p2)) by
A93,
XBOOLE_0:def 4;
(p00
`2 )
<= (p2
`2 ) by
A22,
A25,
EUCLID: 52;
then
A95: (p
`2 )
<= (p2
`2 ) by
A94,
TOPREAL1: 4;
A96: p
in (
LSeg (p2,p1)) by
A93,
XBOOLE_0:def 4;
then (p2
`2 )
<= (p
`2 ) by
A15,
A22,
A72,
TOPREAL1: 4;
then
A97: (p2
`2 )
= (p
`2 ) by
A95,
XXREAL_0: 1;
(p2
`1 )
<= (p
`1 ) by
A15,
A16,
A22,
A23,
A96,
TOPREAL1: 3;
then (p
`1 )
=
0 by
A15,
A16,
A22,
A23,
A96,
TOPREAL1: 3;
then p
=
|[
0 , (p2
`2 )]| by
A97,
EUCLID: 53
.= p2 by
A22,
A23,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A98: ((
LSeg (p01,p1))
/\ L3)
c=
{p00} by
A3,
Lm22,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
take P1 = (
LSeg (p2,p1)), P2 = ((
LSeg (p2,p00))
\/ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p1))));
A99: p2
in (
LSeg (p2,p00)) by
RLTOPSP1: 68;
p2
in (
LSeg (p2,p1)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p2,p1))
/\ (
LSeg (p2,p00))) by
A99,
XBOOLE_0:def 4;
then
A100:
{p2}
c= ((
LSeg (p2,p1))
/\ (
LSeg (p2,p00))) by
ZFMISC_1: 31;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A101: (L2
/\ (L3
\/ L4))
= (
{}
\/
{p11}) by
Lm2,
TOPREAL1: 18,
XBOOLE_1: 23
.=
{p11};
(L3
\/ L4)
is_an_arc_of (p11,p00) by
Lm4,
Lm8,
TOPREAL1: 12,
TOPREAL1: 16;
then
A102: ((L3
\/ L4)
\/ (
LSeg (p11,p01)))
is_an_arc_of (p01,p00) by
A101,
TOPREAL1: 11;
p00
in (
LSeg (p2,p00)) by
RLTOPSP1: 68;
then p00
in ((
LSeg (p2,p00))
/\ L3) by
Lm21,
XBOOLE_0:def 4;
then
A103:
{p00}
c= ((
LSeg (p2,p00))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))
= (((
LSeg (p01,p1))
/\ (L3
\/ L4))
\/ ((
LSeg (p01,p1))
/\ (
LSeg (p11,p01)))) by
XBOOLE_1: 23
.= ((((
LSeg (p01,p1))
/\ L3)
\/ ((
LSeg (p01,p1))
/\ L4))
\/ ((
LSeg (p01,p1))
/\ (
LSeg (p11,p01)))) by
XBOOLE_1: 23
.= ((
{}
\/ ((
LSeg (p01,p1))
/\ L4))
\/ ((
LSeg (p01,p1))
/\ (
LSeg (p11,p01)))) by
A98,
A90,
ZFMISC_1: 33
.=
{p01} by
A14,
A10,
A12,
XBOOLE_0:def 10;
then
A104: (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p1)))
is_an_arc_of (p1,p00) by
A102,
TOPREAL1: 11;
((((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p1)))
/\ (
LSeg (p00,p2)))
= (((
LSeg (p2,p00))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)))) by
XBOOLE_1: 23
.= ((((
LSeg (p2,p00))
/\ (L3
\/ L4))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)))) by
XBOOLE_1: 23
.= (((((
LSeg (p2,p00))
/\ L3)
\/ ((
LSeg (p2,p00))
/\ L4))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)))) by
XBOOLE_1: 23
.= (((((
LSeg (p2,p00))
/\ L3)
\/
{} )
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p00))
/\ (
LSeg (p01,p1)))) by
A79,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26
.= (((
LSeg (p2,p00))
/\ L3)
\/
{} ) by
A80,
A91,
A87,
ZFMISC_1: 33
.=
{p00} by
A103,
A88,
XBOOLE_0:def 10;
hence P2
is_an_arc_of (p1,p2) by
A104,
TOPREAL1: 10;
(((
LSeg (p01,p1))
\/ (
LSeg (p1,p2)))
\/ (
LSeg (p2,p00)))
= L1 by
A3,
A19,
TOPREAL1: 7;
hence
R^2-unit_square
= ((((
LSeg (p2,p1))
\/ (
LSeg (p01,p1)))
\/ (
LSeg (p2,p00)))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01)))) by
TOPREAL1:def 2,
XBOOLE_1: 4
.= (((
LSeg (p2,p1))
\/ ((
LSeg (p2,p00))
\/ (
LSeg (p01,p1))))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01)))) by
XBOOLE_1: 4
.= ((
LSeg (p2,p1))
\/ (((
LSeg (p2,p00))
\/ (
LSeg (p01,p1)))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A105: p1
in (
LSeg (p01,p1)) by
RLTOPSP1: 68;
p1
in (
LSeg (p2,p1)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))) by
A105,
XBOOLE_0:def 4;
then
A106:
{p1}
c= ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))) by
ZFMISC_1: 31;
A107: (P1
/\ P2)
= (((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)))
\/ ((
LSeg (p2,p1))
/\ (((L3
\/ L4)
\/ (
LSeg (p11,p01)))
\/ (
LSeg (p01,p1))))) by
XBOOLE_1: 23
.= (((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)))
\/ (((
LSeg (p2,p1))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))))) by
XBOOLE_1: 23
.= (((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)))
\/ ((((
LSeg (p2,p1))
/\ (L3
\/ L4))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))))) by
XBOOLE_1: 23
.= (((
LSeg (p2,p1))
/\ (
LSeg (p2,p00)))
\/ (((((
LSeg (p2,p1))
/\ L3)
\/ ((
LSeg (p2,p1))
/\ L4))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))))) by
XBOOLE_1: 23
.= (
{p2}
\/ (((((
LSeg (p2,p1))
/\ L3)
\/ ((
LSeg (p2,p1))
/\ L4))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))))) by
A100,
A92,
XBOOLE_0:def 10
.= (
{p2}
\/ (((((
LSeg (p2,p1))
/\ L3)
\/
{} )
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p11,p01))))
\/ ((
LSeg (p2,p1))
/\ (
LSeg (p01,p1))))) by
A20,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26
.= (
{p2}
\/ ((((
LSeg (p2,p1))
/\ L3)
\/ ((
LSeg (p2,p1))
/\ L2))
\/
{p1})) by
A106,
A73,
XBOOLE_0:def 10
.= (
{p2}
\/ (((
LSeg (p2,p1))
/\ L3)
\/ (((
LSeg (p2,p1))
/\ L2)
\/
{p1}))) by
XBOOLE_1: 4
.= ((
{p2}
\/ ((
LSeg (p2,p1))
/\ L3))
\/ (((
LSeg (p2,p1))
/\ L2)
\/
{p1})) by
XBOOLE_1: 4;
A108:
now
per cases ;
suppose
A109: p2
= p00;
p2
in (
LSeg (p2,p1)) by
RLTOPSP1: 68;
then ((
LSeg (p2,p1))
/\ L3)
<>
{} by
A109,
Lm21,
XBOOLE_0:def 4;
then ((
LSeg (p2,p1))
/\ L3)
=
{p2} by
A85,
A109,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p2}
\/ (((
LSeg (p2,p1))
/\ L2)
\/
{p1})) by
A107;
end;
suppose
A110: p2
<> p00;
now
assume p00
in ((
LSeg (p2,p1))
/\ L3);
then p00
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p2
`2 )
<= (p00
`2 ) by
A15,
A22,
A72,
TOPREAL1: 4;
then (p00
`2 )
= (p2
`2 ) by
A19,
Lm5,
Lm7,
TOPREAL1: 4;
hence contradiction by
A22,
A23,
A110,
Lm5,
EUCLID: 53;
end;
then ((
LSeg (p2,p1))
/\ L3)
<>
{p00} by
ZFMISC_1: 31;
then ((
LSeg (p2,p1))
/\ L3)
=
{} by
A85,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p2}
\/ (((
LSeg (p2,p1))
/\ L2)
\/
{p1})) by
A107;
end;
end;
A111: ((
LSeg (p2,p1))
/\ L2)
c=
{p01} by
A3,
A19,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
now
per cases ;
suppose
A112: p1
<> p01;
now
assume p01
in ((
LSeg (p2,p1))
/\ L2);
then p01
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then
A113: (p01
`2 )
<= (p1
`2 ) by
A15,
A22,
A72,
TOPREAL1: 4;
(p1
`2 )
<= (p01
`2 ) by
A3,
Lm5,
Lm7,
TOPREAL1: 4;
then
A114: (p01
`2 )
= (p1
`2 ) by
A113,
XXREAL_0: 1;
p1
=
|[(p1
`1 ), (p1
`2 )]| by
EUCLID: 53
.=
|[
0 , 1]| by
A15,
A16,
A114,
EUCLID: 52;
hence contradiction by
A112;
end;
then ((
LSeg (p2,p1))
/\ L2)
<>
{p01} by
ZFMISC_1: 31;
then ((
LSeg (p2,p1))
/\ L2)
=
{} by
A111,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A108,
ENUMSET1: 1;
end;
suppose
A115: p1
= p01;
then p01
in (
LSeg (p2,p1)) by
RLTOPSP1: 68;
then ((
LSeg (p2,p1))
/\ L2)
<>
{} by
Lm23,
XBOOLE_0:def 4;
then ((
LSeg (p2,p1))
/\ L2)
=
{p1} by
A111,
A115,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A108,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
end;
hence thesis;
end;
suppose
A116: p2
in L2;
then
A117: (
LSeg (p01,p2))
c= L2 by
Lm23,
TOPREAL1: 6;
(
LSeg (p1,p01))
c= L1 by
A3,
Lm22,
TOPREAL1: 6;
then
A118: ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
c= (L1
/\ L2) by
A117,
XBOOLE_1: 27;
take P1 = ((
LSeg (p1,p01))
\/ (
LSeg (p01,p2))), P2 = ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2))));
A119: p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A120: p11
in (L4
/\ (
LSeg (p11,p2))) by
Lm27,
XBOOLE_0:def 4;
p01
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
<>
{} by
A119,
XBOOLE_0:def 4;
then
A121: ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
=
{p01} by
A118,
TOPREAL1: 15,
ZFMISC_1: 33;
p1
<> p01 or p2
<> p01 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A121,
TOPREAL1: 12;
A122: L1
= ((
LSeg (p1,p01))
\/ (
LSeg (p1,p00))) by
A3,
TOPREAL1: 5;
A123: L4
is_an_arc_of (p10,p11) by
Lm9,
Lm11,
TOPREAL1: 9;
L3
is_an_arc_of (p00,p10) by
Lm4,
Lm8,
TOPREAL1: 9;
then
A124: (L3
\/ L4)
is_an_arc_of (p00,p11) by
A123,
TOPREAL1: 2,
TOPREAL1: 16;
A125: (
LSeg (p11,p2))
c= L2 by
A116,
Lm26,
TOPREAL1: 6;
then
A126: (L4
/\ (
LSeg (p11,p2)))
c= (L4
/\ L2) by
XBOOLE_1: 27;
A127: (L3
/\ (
LSeg (p11,p2)))
=
{} by
A125,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L3
\/ L4)
/\ (
LSeg (p11,p2)))
= ((L3
/\ (
LSeg (p11,p2)))
\/ (L4
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.=
{p11} by
A127,
A126,
A120,
TOPREAL1: 18,
ZFMISC_1: 33;
then
A128: ((L3
\/ L4)
\/ (
LSeg (p11,p2)))
is_an_arc_of (p00,p2) by
A124,
TOPREAL1: 10;
A129: ((
LSeg (p01,p2))
/\ (
LSeg (p11,p2)))
=
{p2} by
A116,
TOPREAL1: 8;
A130: L2
= ((
LSeg (p11,p2))
\/ (
LSeg (p01,p2))) by
A116,
TOPREAL1: 5;
((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
c=
{p01} by
A4,
A125,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A131: ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
=
{p01} or ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
=
{} by
ZFMISC_1: 33;
A132: (
LSeg (p01,p2))
c= L2 by
A116,
Lm23,
TOPREAL1: 6;
then
A133: ((
LSeg (p01,p2))
/\ L3)
=
{} by
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A134: ex q st q
= p2 & (q
`1 )
<= 1 & (q
`1 )
>=
0 & (q
`2 )
= 1 by
A116,
TOPREAL1: 13;
A135:
now
A136: (p2
`1 )
<= (p11
`1 ) by
A134,
EUCLID: 52;
assume
A137: p01
in ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)));
then
A138: p01
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
p01
in (
LSeg (p2,p11)) by
A137,
XBOOLE_0:def 4;
then
A139: (p01
`1 )
= (p2
`1 ) by
A134,
A136,
Lm6,
TOPREAL1: 3;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p01
`2 )
<= (p1
`2 ) by
A138,
TOPREAL1: 4;
then (p01
`2 )
= (p1
`2 ) by
A15,
A17,
Lm7,
XXREAL_0: 1;
then p1
=
|[(p01
`1 ), (p01
`2 )]| by
A15,
A16,
Lm6,
EUCLID: 53
.= p2 by
A134,
A139,
Lm7,
EUCLID: 53;
hence contradiction by
A1;
end;
((
LSeg (p1,p00))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p00))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p00))
/\ L3)
\/ ((
LSeg (p1,p00))
/\ L4)) by
A131,
A135,
XBOOLE_1: 23,
ZFMISC_1: 31
.=
{p00} by
A9,
A5,
A6,
TOPREAL1: 17,
XBOOLE_0:def 10;
hence P2
is_an_arc_of (p1,p2) by
A128,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p01,p2))
\/ ((
LSeg (p1,p01))
\/ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 4
.= ((L1
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2))))
\/ (
LSeg (p01,p2))) by
A122,
XBOOLE_1: 4
.= (L1
\/ (((L3
\/ L4)
\/ (
LSeg (p11,p2)))
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= (L1
\/ (L2
\/ (L3
\/ L4))) by
A130,
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A140:
{p1}
= ((
LSeg (p1,p01))
/\ (
LSeg (p1,p00))) by
A3,
TOPREAL1: 8;
A141: (P1
/\ P2)
= (((
LSeg (p1,p01))
/\ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p01))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p1,p01))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
A140,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p00))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p01,p2))
/\ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/ (((
LSeg (p01,p2))
/\ (L3
\/ L4))
\/
{p2}))) by
A129,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/ ((((
LSeg (p01,p2))
/\ L3)
\/ ((
LSeg (p01,p2))
/\ L4))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2}))) by
A14,
A133;
A142:
now
per cases ;
suppose
A143: p2
= p11;
then
A144: not p2
in (
LSeg (p1,p01)) by
A13,
Lm4,
Lm6,
Lm10,
TOPREAL1: 3;
((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
= ((
LSeg (p1,p01))
/\
{p2}) by
A143,
RLTOPSP1: 70
.=
{} by
A144,
Lm1;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/
{p2})) by
A141,
A143,
TOPREAL1: 18;
end;
suppose
A145: p2
<> p11 & p2
<> p01;
now
assume p01
in ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)));
then
A146: p01
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A134,
EUCLID: 52;
then (p2
`1 )
=
0 by
A134,
A146,
Lm6,
TOPREAL1: 3;
hence contradiction by
A134,
A145,
EUCLID: 53;
end;
then
A147:
{p01}
<> ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
c=
{p01} by
A13,
A125,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A148: ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
=
{} by
A147,
ZFMISC_1: 33;
now
assume p11
in ((
LSeg (p01,p2))
/\ L4);
then
A149: p11
in (
LSeg (p01,p2)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A134,
EUCLID: 52;
then (p11
`1 )
<= (p2
`1 ) by
A149,
TOPREAL1: 3;
then (p2
`1 )
= (p11
`1 ) by
A134,
Lm10,
XXREAL_0: 1;
hence contradiction by
A134,
A145,
Lm10,
EUCLID: 53;
end;
then
A150:
{p11}
<> ((
LSeg (p01,p2))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ L4)
c=
{p11} by
A132,
TOPREAL1: 18,
XBOOLE_1: 27;
then ((
LSeg (p01,p2))
/\ L4)
=
{} by
A150,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/
{p2})) by
A141,
A148;
end;
suppose
A151: p2
= p01;
then p2
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
then
A152: ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
<>
{} by
A151,
Lm23,
XBOOLE_0:def 4;
((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
c=
{p2} by
A13,
A151,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A153: ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
=
{p2} by
A152,
ZFMISC_1: 33;
((
LSeg (p01,p2))
/\ L4)
= (
{p01}
/\ L4) by
A151,
RLTOPSP1: 70
.=
{} by
Lm1,
Lm15;
hence (P1
/\ P2)
= (((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/
{p2})
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/
{p2})) by
A141,
A153,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/ ((((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/
{p2})
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/ (
{p2}
\/
{p2}))) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ L3))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
\/
{p2}));
end;
end;
now
per cases ;
suppose
A154: p1
= p01;
then p1
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A155: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
<>
{} by
A154,
Lm22,
XBOOLE_0:def 4;
((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
c=
{p1} by
A132,
A154,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A156: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
=
{p1} by
A155,
ZFMISC_1: 33;
((
LSeg (p1,p01))
/\ L3)
= (
{p1}
/\ L3) by
A154,
RLTOPSP1: 70
.=
{} by
A154,
Lm1,
Lm14;
hence (P1
/\ P2)
= ((
{p1}
\/
{p1})
\/
{p2}) by
A142,
A156,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A157: p1
= p00;
A158: not p00
in (
LSeg (p01,p2)) by
A132,
Lm5,
Lm7,
Lm11,
TOPREAL1: 4;
((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
= ((
LSeg (p01,p2))
/\
{p00}) by
A157,
RLTOPSP1: 70
.=
{} by
A158,
Lm1;
hence thesis by
A142,
A157,
ENUMSET1: 1,
TOPREAL1: 17;
end;
suppose
A159: p1
<> p00 & p1
<> p01;
now
assume p01
in ((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)));
then
A160: p01
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p01
`2 )
<= (p1
`2 ) by
A160,
TOPREAL1: 4;
then (p1
`2 )
= 1 by
A15,
A17,
Lm7,
XXREAL_0: 1;
hence contradiction by
A15,
A16,
A159,
EUCLID: 53;
end;
then
A161:
{p01}
<> ((
LSeg (p01,p2))
/\ (
LSeg (p1,p00))) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
c=
{p01} by
A4,
A132,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A162: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p00)))
=
{} by
A161,
ZFMISC_1: 33;
now
assume p00
in ((
LSeg (p1,p01))
/\ L3);
then
A163: p00
in (
LSeg (p1,p01)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
then (p1
`2 )
=
0 by
A15,
A18,
A163,
Lm5,
TOPREAL1: 4;
hence contradiction by
A15,
A16,
A159,
EUCLID: 53;
end;
then
A164:
{p00}
<> ((
LSeg (p1,p01))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ L3)
c=
{p00} by
A13,
TOPREAL1: 17,
XBOOLE_1: 27;
then ((
LSeg (p1,p01))
/\ L3)
=
{} by
A164,
ZFMISC_1: 33;
hence thesis by
A142,
A162,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A165: p2
in L3;
then
A166: (
LSeg (p00,p2))
c= L3 by
Lm21,
TOPREAL1: 6;
(
LSeg (p1,p00))
c= L1 by
A3,
Lm20,
TOPREAL1: 6;
then
A167: ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2)))
c= (L1
/\ L3) by
A166,
XBOOLE_1: 27;
take P1 = ((
LSeg (p1,p00))
\/ (
LSeg (p00,p2))), P2 = ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2))));
A168: p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A169: p10
in (L4
/\ (
LSeg (p10,p2))) by
Lm25,
XBOOLE_0:def 4;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2)))
<>
{} by
A168,
XBOOLE_0:def 4;
then
A170: ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2)))
=
{p00} by
A167,
TOPREAL1: 17,
ZFMISC_1: 33;
p1
<> p00 or p00
<> p2 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A170,
TOPREAL1: 12;
A171: L1
= ((
LSeg (p1,p00))
\/ (
LSeg (p1,p01))) by
A3,
TOPREAL1: 5;
A172: L4
is_an_arc_of (p11,p10) by
Lm9,
Lm11,
TOPREAL1: 9;
L2
is_an_arc_of (p01,p11) by
Lm6,
Lm10,
TOPREAL1: 9;
then
A173: (L2
\/ L4)
is_an_arc_of (p01,p10) by
A172,
TOPREAL1: 2,
TOPREAL1: 18;
A174: (
LSeg (p10,p2))
c= L3 by
A165,
Lm24,
TOPREAL1: 6;
then
A175: (L4
/\ (
LSeg (p10,p2)))
c= (L4
/\ L3) by
XBOOLE_1: 27;
A176: (L2
/\ (
LSeg (p10,p2)))
=
{} by
A174,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L2
\/ L4)
/\ (
LSeg (p10,p2)))
= ((L2
/\ (
LSeg (p10,p2)))
\/ (L4
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.=
{p10} by
A176,
A175,
A169,
TOPREAL1: 16,
ZFMISC_1: 33;
then
A177: ((L2
\/ L4)
\/ (
LSeg (p10,p2)))
is_an_arc_of (p01,p2) by
A173,
TOPREAL1: 10;
A178: ((
LSeg (p00,p2))
/\ (
LSeg (p10,p2)))
=
{p2} by
A165,
TOPREAL1: 8;
A179: L3
= ((
LSeg (p10,p2))
\/ (
LSeg (p00,p2))) by
A165,
TOPREAL1: 5;
((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))
c=
{p00} by
A13,
A174,
TOPREAL1: 17,
XBOOLE_1: 27;
then
A180: ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))
=
{p00} or ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))
=
{} by
ZFMISC_1: 33;
A181: (
LSeg (p00,p2))
c= L3 by
A165,
Lm21,
TOPREAL1: 6;
then
A182: ((
LSeg (p00,p2))
/\ L2)
=
{} by
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A183: ex q st q
= p2 & (q
`1 )
<= 1 & (q
`1 )
>=
0 & (q
`2 )
=
0 by
A165,
TOPREAL1: 13;
A184:
now
A185: (p2
`1 )
<= (p10
`1 ) by
A183,
EUCLID: 52;
assume
A186: p00
in ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)));
then
A187: p00
in (
LSeg (p1,p01)) by
XBOOLE_0:def 4;
p00
in (
LSeg (p2,p10)) by
A186,
XBOOLE_0:def 4;
then
A188: (p00
`1 )
= (p2
`1 ) by
A183,
A185,
Lm4,
TOPREAL1: 3;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
then (p00
`2 )
= (p1
`2 ) by
A15,
A18,
A187,
Lm5,
TOPREAL1: 4;
then p1
=
|[(p00
`1 ), (p00
`2 )]| by
A15,
A16,
Lm4,
EUCLID: 53
.= p2 by
A183,
A188,
Lm5,
EUCLID: 53;
hence contradiction by
A1;
end;
((
LSeg (p1,p01))
/\ ((L2
\/ L4)
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p01))
/\ (L2
\/ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p01))
/\ L2)
\/ ((
LSeg (p01,p1))
/\ L4)) by
A180,
A184,
XBOOLE_1: 23,
ZFMISC_1: 31
.=
{p01} by
A14,
A10,
A12,
XBOOLE_0:def 10;
hence P2
is_an_arc_of (p1,p2) by
A177,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p00,p2))
\/ ((
LSeg (p1,p00))
\/ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 4
.= ((L1
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2))))
\/ (
LSeg (p00,p2))) by
A171,
XBOOLE_1: 4
.= (L1
\/ (((L2
\/ L4)
\/ (
LSeg (p10,p2)))
\/ (
LSeg (p00,p2)))) by
XBOOLE_1: 4
.= (L1
\/ ((L2
\/ L4)
\/ ((
LSeg (p10,p2))
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 4
.= (L1
\/ (L2
\/ (L3
\/ L4))) by
A179,
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A189:
{p1}
= ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01))) by
A3,
TOPREAL1: 8;
A190: (P1
/\ P2)
= (((
LSeg (p1,p00))
/\ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p1,p00))
/\ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ (L2
\/ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
A189,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p00))
/\ L2)
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p01))
\/ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p00))
/\ L2)
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p00,p2))
/\ ((L2
\/ L4)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p00))
/\ L2)
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/ (((
LSeg (p00,p2))
/\ (L2
\/ L4))
\/
{p2}))) by
A178,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p00))
/\ L2)
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/ ((((
LSeg (p00,p2))
/\ L2)
\/ ((
LSeg (p00,p2))
/\ L4))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ L2)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/ (((
LSeg (p00,p2))
/\ L4)
\/
{p2}))) by
A9,
A182;
A191:
now
per cases ;
suppose
A192: p2
= p10;
then not p2
in (
LSeg (p1,p00)) by
A4,
Lm4,
Lm6,
Lm8,
TOPREAL1: 3;
then
A193: (
LSeg (p1,p00))
misses
{p2} by
ZFMISC_1: 50;
((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
= ((
LSeg (p1,p00))
/\
{p2}) by
A192,
RLTOPSP1: 70
.=
{} by
A193,
XBOOLE_0:def 7;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A190,
A192,
TOPREAL1: 16;
end;
suppose
A194: p2
<> p10 & p2
<> p00;
now
assume p00
in ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)));
then
A195: p00
in (
LSeg (p2,p10)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A183,
EUCLID: 52;
then (p2
`1 )
=
0 by
A183,
A195,
Lm4,
TOPREAL1: 3;
hence contradiction by
A183,
A194,
EUCLID: 53;
end;
then
A196:
{p00}
<> ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
c=
{p00} by
A4,
A174,
TOPREAL1: 17,
XBOOLE_1: 27;
then
A197: ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
=
{} by
A196,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p00,p2))
/\ L4);
then
A198: p10
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A183,
EUCLID: 52;
then (p10
`1 )
<= (p2
`1 ) by
A198,
TOPREAL1: 3;
then (p2
`1 )
= (p10
`1 ) by
A183,
Lm8,
XXREAL_0: 1;
hence contradiction by
A183,
A194,
Lm8,
EUCLID: 53;
end;
then
A199:
{p10}
<> ((
LSeg (p00,p2))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p00,p2))
/\ L4)
c=
{p10} by
A181,
TOPREAL1: 16,
XBOOLE_1: 27;
then ((
LSeg (p00,p2))
/\ L4)
=
{} by
A199,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A190,
A197;
end;
suppose
A200: p2
= p00;
then p2
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then
A201: ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
<>
{} by
A200,
Lm21,
XBOOLE_0:def 4;
((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
c=
{p2} by
A4,
A200,
TOPREAL1: 17,
XBOOLE_1: 27;
then
A202: ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
=
{p2} by
A201,
ZFMISC_1: 33;
((
LSeg (p00,p2))
/\ L4)
= (
{p00}
/\ L4) by
A200,
RLTOPSP1: 70
.=
{} by
Lm1,
Lm12;
hence (P1
/\ P2)
= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/
{p2})
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A190,
A202,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/ (
{p2}
\/
{p2}))) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
\/
{p2}));
end;
end;
now
per cases ;
suppose
A203: p1
= p01;
then
A204: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
= ((
LSeg (p00,p2))
/\
{p1}) by
RLTOPSP1: 70;
not p1
in (
LSeg (p00,p2)) by
A181,
A203,
Lm5,
Lm7,
Lm9,
TOPREAL1: 4;
then ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A204,
Lm1;
hence thesis by
A191,
A203,
ENUMSET1: 1,
TOPREAL1: 15;
end;
suppose
A205: p1
= p00;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A206: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
<>
{} by
A205,
Lm20,
XBOOLE_0:def 4;
((
LSeg (p1,p00))
/\ L2)
= (
{p1}
/\ L2) by
A205,
RLTOPSP1: 70;
then
A207: ((
LSeg (p1,p00))
/\ L2)
=
{} by
A205,
Lm1,
Lm13;
((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
c= (L3
/\ L1) by
A165,
A205,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
=
{p1} by
A205,
A206,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/
{p1})
\/
{p2}) by
A191,
A207,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A208: p1
<> p00 & p1
<> p01;
now
assume p00
in ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)));
then
A209: p00
in (
LSeg (p1,p01)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p01
`2 ) by
A15,
A17,
EUCLID: 52;
then (p1
`2 )
=
0 by
A15,
A18,
A209,
Lm5,
TOPREAL1: 4;
hence contradiction by
A15,
A16,
A208,
EUCLID: 53;
end;
then
A210:
{p00}
<> ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
c= (L3
/\ L1) by
A13,
A181,
XBOOLE_1: 27;
then
A211: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A210,
TOPREAL1: 17,
ZFMISC_1: 33;
now
assume p01
in ((
LSeg (p1,p00))
/\ L2);
then
A212: p01
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p01
`2 )
<= (p1
`2 ) by
A212,
TOPREAL1: 4;
then (p1
`2 )
= 1 by
A15,
A17,
Lm7,
XXREAL_0: 1;
hence contradiction by
A15,
A16,
A208,
EUCLID: 53;
end;
then
A213:
{p01}
<> ((
LSeg (p1,p00))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ L2)
c=
{p01} by
A4,
TOPREAL1: 15,
XBOOLE_1: 27;
then ((
LSeg (p1,p00))
/\ L2)
=
{} by
A213,
ZFMISC_1: 33;
hence thesis by
A191,
A211,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A214: p2
in L4;
now
let a be
object;
assume
A215: a
in ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
a
in (
LSeg (p1,p01)) by
A215,
XBOOLE_0:def 4;
then
A216: (p1
`2 )
<= (p
`2 ) by
A15,
A17,
Lm7,
TOPREAL1: 4;
A217: a
in (
LSeg (p00,p1)) by
A215,
XBOOLE_0:def 4;
then (p
`2 )
<= (p1
`2 ) by
A15,
A18,
Lm5,
TOPREAL1: 4;
then
A218: (p
`2 )
= (p1
`2 ) by
A216,
XXREAL_0: 1;
(p
`1 )
<= (p1
`1 ) by
A15,
A16,
A217,
Lm4,
TOPREAL1: 3;
then (p
`1 )
= (p1
`1 ) by
A15,
A16,
A217,
Lm4,
TOPREAL1: 3;
then a
=
|[(p1
`1 ), (p1
`2 )]| by
A218,
EUCLID: 53
.= p1 by
EUCLID: 53;
hence a
in
{p1} by
TARSKI:def 1;
end;
then
A219: ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01)))
c=
{p1};
A220: p2
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2))) by
A220,
XBOOLE_0:def 4;
then
A221:
{p2}
c= ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
A222: ex q st q
= p2 & (q
`1 )
= 1 & (q
`2 )
<= 1 & (q
`2 )
>=
0 by
A214,
TOPREAL1: 13;
now
let a be
object;
assume
A223: a
in ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A224: a
in (
LSeg (p10,p2)) by
A223,
XBOOLE_0:def 4;
then
A225: (p2
`1 )
<= (p
`1 ) by
A222,
Lm8,
TOPREAL1: 3;
a
in (
LSeg (p2,p11)) by
A223,
XBOOLE_0:def 4;
then
A226: (p2
`2 )
<= (p
`2 ) by
A222,
Lm11,
TOPREAL1: 4;
(p
`1 )
<= (p2
`1 ) by
A222,
A224,
Lm8,
TOPREAL1: 3;
then
A227: (p
`1 )
= (p2
`1 ) by
A225,
XXREAL_0: 1;
(p
`2 )
<= (p2
`2 ) by
A222,
A224,
Lm9,
TOPREAL1: 4;
then (p
`2 )
= (p2
`2 ) by
A226,
XXREAL_0: 1;
then a
=
|[(p2
`1 ), (p2
`2 )]| by
A227,
EUCLID: 53
.= p2 by
EUCLID: 53;
hence a
in
{p2} by
TARSKI:def 1;
end;
then
A228: ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2)))
c=
{p2};
(
LSeg (p10,p2))
c= L4 by
A214,
Lm25,
TOPREAL1: 6;
then
A229: (L3
/\ (
LSeg (p10,p2)))
c=
{p10} by
TOPREAL1: 16,
XBOOLE_1: 27;
take P1 = (((
LSeg (p1,p00))
\/ L3)
\/ (
LSeg (p10,p2))), P2 = (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2)));
A230: p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
p10
in L3 by
RLTOPSP1: 68;
then (L3
/\ (
LSeg (p10,p2)))
<>
{} by
A230,
XBOOLE_0:def 4;
then (L3
/\ (
LSeg (p10,p2)))
=
{p10} by
A229,
ZFMISC_1: 33;
then
A231: (L3
\/ (
LSeg (p10,p2)))
is_an_arc_of (p00,p2) by
Lm4,
Lm8,
TOPREAL1: 12;
(
LSeg (p10,p2))
c= L4 by
A214,
Lm25,
TOPREAL1: 6;
then
A232: ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
=
{} by
A4,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
((
LSeg (p1,p00))
/\ (L3
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p00))
/\ L3)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.=
{p00} by
A5,
A6,
A232,
TOPREAL1: 17,
XBOOLE_0:def 10;
then ((
LSeg (p1,p00))
\/ (L3
\/ (
LSeg (p10,p2))))
is_an_arc_of (p1,p2) by
A231,
TOPREAL1: 11;
hence P1
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A233: (L2
/\ (
LSeg (p11,p2)))
<>
{} by
Lm26,
XBOOLE_0:def 4;
A234: (
LSeg (p11,p2))
c= L4 by
A214,
Lm27,
TOPREAL1: 6;
then
A235: ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
=
{} by
A13,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
(L2
/\ (
LSeg (p11,p2)))
c=
{p11} by
A234,
TOPREAL1: 18,
XBOOLE_1: 27;
then (L2
/\ (
LSeg (p11,p2)))
=
{p11} by
A233,
ZFMISC_1: 33;
then
A236: (L2
\/ (
LSeg (p11,p2)))
is_an_arc_of (p01,p2) by
Lm6,
Lm10,
TOPREAL1: 12;
((
LSeg (p1,p01))
/\ (L2
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p01))
/\ L2)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.=
{p01} by
A10,
A12,
A235,
XBOOLE_0:def 10;
then ((
LSeg (p1,p01))
\/ (L2
\/ (
LSeg (p11,p2))))
is_an_arc_of (p1,p2) by
A236,
TOPREAL1: 11;
hence P2
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
thus
R^2-unit_square
= ((((
LSeg (p1,p00))
\/ (
LSeg (p1,p01)))
\/ L2)
\/ (L3
\/ L4)) by
A3,
TOPREAL1: 5,
TOPREAL1:def 2
.= (((
LSeg (p1,p00))
\/ ((
LSeg (p1,p01))
\/ L2))
\/ (L3
\/ L4)) by
XBOOLE_1: 4
.= ((
LSeg (p1,p00))
\/ (((
LSeg (p1,p01))
\/ L2)
\/ (L3
\/ L4))) by
XBOOLE_1: 4
.= ((
LSeg (p1,p00))
\/ (L3
\/ (((
LSeg (p1,p01))
\/ L2)
\/ L4))) by
XBOOLE_1: 4
.= (((
LSeg (p1,p00))
\/ L3)
\/ (((
LSeg (p1,p01))
\/ L2)
\/ L4)) by
XBOOLE_1: 4
.= (((
LSeg (p1,p00))
\/ L3)
\/ (((
LSeg (p1,p01))
\/ L2)
\/ ((
LSeg (p11,p2))
\/ (
LSeg (p10,p2))))) by
A214,
TOPREAL1: 5
.= (((
LSeg (p1,p00))
\/ L3)
\/ ((
LSeg (p10,p2))
\/ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A237: p1
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
p1
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01))) by
A237,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
then
A238: ((
LSeg (p1,p00))
/\ (
LSeg (p1,p01)))
=
{p1} by
A219,
XBOOLE_0:def 10;
A239: ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
=
{} by
A4,
A234,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
A240: (
LSeg (p10,p2))
c= L4 by
A214,
Lm25,
TOPREAL1: 6;
then
A241: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A13,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
A242: (P1
/\ P2)
= ((((
LSeg (p1,p00))
\/ L3)
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))
\/ (L3
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p00))
/\ ((
LSeg (p1,p01))
\/ L2))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2))))
\/ (L3
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p00))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (L3
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
A239,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ ((
LSeg (p1,p01))
\/ L2))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
A238,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((L3
/\ (
LSeg (p1,p01)))
\/ (L3
/\ L2))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ (
LSeg (p1,p01)))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ ((
LSeg (p1,p01))
\/ L2))
\/ ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2))))) by
Lm2,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ (
LSeg (p1,p01)))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ ((
LSeg (p1,p01))
\/ L2))
\/
{p2})) by
A221,
A228,
XBOOLE_0:def 10
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ (
LSeg (p1,p01)))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ ((((
LSeg (p10,p2))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p10,p2))
/\ L2))
\/
{p2})) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ (
LSeg (p1,p01)))
\/ (L3
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2})) by
A241;
A243:
now
per cases ;
suppose
A244: p2
= p11;
then (L3
/\ (
LSeg (p11,p2)))
= (L3
/\
{p11}) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm19;
hence (P1
/\ P2)
= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (L3
/\ (
LSeg (p1,p01))))
\/
{p2}) by
A242,
A244,
TOPREAL1: 18;
end;
suppose
A245: p2
= p10;
then ((
LSeg (p10,p2))
/\ L2)
= (
{p10}
/\ L2) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm17;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (((L3
/\ (
LSeg (p1,p01)))
\/
{p2})
\/
{p2})) by
A242,
A245,
TOPREAL1: 16,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((L3
/\ (
LSeg (p1,p01)))
\/ (
{p2}
\/
{p2}))) by
XBOOLE_1: 4
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (L3
/\ (
LSeg (p1,p01))))
\/
{p2}) by
XBOOLE_1: 4;
end;
suppose
A246: p2
<> p10 & p2
<> p11;
now
assume p11
in ((
LSeg (p10,p2))
/\ L2);
then
A247: p11
in (
LSeg (p10,p2)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A222,
EUCLID: 52;
then (p11
`2 )
<= (p2
`2 ) by
A247,
TOPREAL1: 4;
then (p11
`2 )
= (p2
`2 ) by
A222,
Lm11,
XXREAL_0: 1;
then p2
=
|[(p11
`1 ), (p11
`2 )]| by
A222,
Lm10,
EUCLID: 53
.= p11 by
EUCLID: 53;
hence contradiction by
A246;
end;
then
A248:
{p11}
<> ((
LSeg (p10,p2))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ L2)
c= (L4
/\ L2) by
A240,
XBOOLE_1: 27;
then
A249: ((
LSeg (p10,p2))
/\ L2)
=
{} by
A248,
TOPREAL1: 18,
ZFMISC_1: 33;
now
assume p10
in (L3
/\ (
LSeg (p11,p2)));
then
A250: p10
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A222,
EUCLID: 52;
then (p2
`2 )
= (p10
`2 ) by
A222,
A250,
Lm9,
TOPREAL1: 4;
then p2
=
|[(p10
`1 ), (p10
`2 )]| by
A222,
Lm8,
EUCLID: 53
.= p10 by
EUCLID: 53;
hence contradiction by
A246;
end;
then
A251: (L3
/\ (
LSeg (p11,p2)))
<>
{p10} by
ZFMISC_1: 31;
(L3
/\ (
LSeg (p11,p2)))
c=
{p10} by
A234,
TOPREAL1: 16,
XBOOLE_1: 27;
then (L3
/\ (
LSeg (p11,p2)))
=
{} by
A251,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L2))
\/ (L3
/\ (
LSeg (p1,p01))))
\/
{p2}) by
A242,
A249;
end;
end;
now
per cases ;
suppose
A252: p1
= p01;
then (L3
/\ (
LSeg (p1,p01)))
= (L3
/\
{p01}) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm14;
hence thesis by
A243,
A252,
ENUMSET1: 1,
TOPREAL1: 15;
end;
suppose
A253: p1
<> p01 & p1
<> p00;
now
assume p01
in ((
LSeg (p1,p00))
/\ L2);
then
A254: p01
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p1
`2 ) by
A15,
A18,
EUCLID: 52;
then (p01
`2 )
<= (p1
`2 ) by
A254,
TOPREAL1: 4;
then (p1
`2 )
= (p01
`2 ) by
A15,
A17,
Lm7,
XXREAL_0: 1;
then p1
=
|[(p01
`1 ), (p01
`2 )]| by
A15,
A16,
Lm6,
EUCLID: 53
.= p01 by
EUCLID: 53;
hence contradiction by
A253;
end;
then
A255:
{p01}
<> ((
LSeg (p1,p00))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ L2)
c=
{p01} by
A4,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A256: ((
LSeg (p1,p00))
/\ L2)
=
{} by
A255,
ZFMISC_1: 33;
now
assume p00
in (L3
/\ (
LSeg (p1,p01)));
then p00
in (
LSeg (p1,p01)) by
XBOOLE_0:def 4;
then (p1
`2 )
= (p00
`2 ) by
A15,
A17,
A18,
Lm5,
Lm7,
TOPREAL1: 4;
then p1
=
|[(p00
`1 ), (p00
`2 )]| by
A15,
A16,
Lm4,
EUCLID: 53
.= p00 by
EUCLID: 53;
hence contradiction by
A253;
end;
then
A257:
{p00}
<> (L3
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
(L3
/\ (
LSeg (p1,p01)))
c= (L3
/\ L1) by
A13,
XBOOLE_1: 27;
then (L3
/\ (
LSeg (p1,p01)))
=
{} by
A257,
TOPREAL1: 17,
ZFMISC_1: 33;
hence thesis by
A243,
A256,
ENUMSET1: 1;
end;
suppose
A258: p1
= p00;
then ((
LSeg (p1,p00))
/\ L2)
= (
{p00}
/\ L2) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm13;
hence thesis by
A243,
A258,
ENUMSET1: 1,
TOPREAL1: 17;
end;
end;
hence thesis;
end;
end;
Lm31: p1
<> p2 & p2
in
R^2-unit_square & p1
in (
LSeg (p01,p11)) implies ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) &
R^2-unit_square
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume that
A1: p1
<> p2 and
A2: p2
in
R^2-unit_square and
A3: p1
in (
LSeg (p01,p11));
A4: p2
in (L1
\/ L2) or p2
in (L3
\/ L4) by
A2,
TOPREAL1:def 2,
XBOOLE_0:def 3;
A5: ((
LSeg (p01,p1))
/\ L1)
c= (L2
/\ L1) by
A3,
Lm23,
TOPREAL1: 6,
XBOOLE_1: 26;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A6: ((
LSeg (p1,p11))
/\ L4)
<>
{} by
Lm27,
XBOOLE_0:def 4;
p01
in (
LSeg (p01,p1)) by
RLTOPSP1: 68;
then
A7: ((
LSeg (p01,p1))
/\ L1)
<>
{} by
Lm22,
XBOOLE_0:def 4;
A8: (
LSeg (p1,p11))
c= L2 by
A3,
Lm26,
TOPREAL1: 6;
then
A9: ((
LSeg (p1,p11))
/\ L3)
=
{} by
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
A10: ((
LSeg (p1,p11))
/\ L4)
c=
{p11} by
A3,
Lm26,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
A11: (
LSeg (p01,p1))
c= L2 by
A3,
Lm23,
TOPREAL1: 6;
then
A12: ((
LSeg (p1,p01))
/\ L3)
=
{} by
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
consider q1 such that
A13: q1
= p1 and
A14: (q1
`1 )
<= 1 and
A15: (q1
`1 )
>=
0 and
A16: (q1
`2 )
= 1 by
A3,
TOPREAL1: 13;
per cases by
A4,
XBOOLE_0:def 3;
suppose
A17: p2
in L1;
then
A18: (
LSeg (p01,p2))
c= L1 by
Lm22,
TOPREAL1: 6;
(
LSeg (p1,p01))
c= L2 by
A3,
Lm23,
TOPREAL1: 6;
then
A19: ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
c= (L2
/\ L1) by
A18,
XBOOLE_1: 27;
take P1 = ((
LSeg (p1,p01))
\/ (
LSeg (p01,p2))), P2 = ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2))));
A20: p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A21:
{}
<> (L3
/\ (
LSeg (p00,p2))) by
Lm21,
XBOOLE_0:def 4;
p01
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
<>
{} by
A20,
XBOOLE_0:def 4;
then
A22: ((
LSeg (p1,p01))
/\ (
LSeg (p01,p2)))
=
{p01} by
A19,
TOPREAL1: 15,
ZFMISC_1: 33;
p1
<> p01 or p2
<> p01 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A22,
TOPREAL1: 12;
A23: ((
LSeg (p1,p01))
\/ (
LSeg (p1,p11)))
= L2 by
A3,
TOPREAL1: 5;
A24: L4
is_an_arc_of (p11,p10) by
Lm9,
Lm11,
TOPREAL1: 9;
L3
is_an_arc_of (p10,p00) by
Lm4,
Lm8,
TOPREAL1: 9;
then
A25: (L3
\/ L4)
is_an_arc_of (p11,p00) by
A24,
TOPREAL1: 2,
TOPREAL1: 16;
A26: (L3
/\ (
LSeg (p00,p2)))
c=
{p00} by
A17,
Lm20,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
A27: ((
LSeg (p00,p2))
\/ (
LSeg (p01,p2)))
= L1 by
A17,
TOPREAL1: 5;
A28: ((
LSeg (p01,p2))
/\ (
LSeg (p00,p2)))
=
{p2} by
A17,
TOPREAL1: 8;
A29: (
LSeg (p00,p2))
c= L1 by
A17,
Lm20,
TOPREAL1: 6;
then
A30: (L4
/\ (
LSeg (p00,p2)))
=
{} by
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
A31: ex q2 st q2
= p2 & (q2
`1 )
=
0 & (q2
`2 )
<= 1 & (q2
`2 )
>=
0 by
A17,
TOPREAL1: 13;
A32:
now
A33: (p00
`2 )
<= (p2
`2 ) by
A31,
EUCLID: 52;
assume
A34: p01
in ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)));
then
A35: p01
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
p01
in (
LSeg (p00,p2)) by
A34,
XBOOLE_0:def 4;
then (p01
`2 )
<= (p2
`2 ) by
A33,
TOPREAL1: 4;
then
A36: (p01
`2 )
= (p2
`2 ) by
A31,
Lm7,
XXREAL_0: 1;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then (p01
`1 )
= (p1
`1 ) by
A13,
A15,
A35,
Lm6,
TOPREAL1: 3;
then p1
=
|[(p01
`1 ), (p01
`2 )]| by
A13,
A16,
Lm7,
EUCLID: 53
.= p2 by
A31,
A36,
Lm6,
EUCLID: 53;
hence contradiction by
A1;
end;
((L3
\/ L4)
/\ (
LSeg (p00,p2)))
= ((L3
/\ (
LSeg (p00,p2)))
\/ (L4
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.=
{p00} by
A26,
A21,
A30,
ZFMISC_1: 33;
then
A37: ((L3
\/ L4)
\/ (
LSeg (p00,p2)))
is_an_arc_of (p11,p2) by
A25,
TOPREAL1: 10;
((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
c= (L2
/\ L1) by
A8,
A29,
XBOOLE_1: 27;
then
A38: ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
=
{p01} or ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
=
{} by
TOPREAL1: 15,
ZFMISC_1: 33;
A39: (
LSeg (p2,p01))
c= L1 by
A17,
Lm22,
TOPREAL1: 6;
then
A40: ((
LSeg (p01,p2))
/\ L4)
=
{} by
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
((
LSeg (p1,p11))
/\ ((L3
\/ L4)
\/ (
LSeg (p00,p2))))
= (((
LSeg (p1,p11))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ L4)) by
A38,
A32,
XBOOLE_1: 23,
ZFMISC_1: 31
.=
{p11} by
A9,
A6,
A10,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A37,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p01,p2))
\/ ((
LSeg (p1,p01))
\/ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 4
.= ((
LSeg (p01,p2))
\/ (L2
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2))))) by
A23,
XBOOLE_1: 4
.= ((
LSeg (p01,p2))
\/ ((L2
\/ (L3
\/ L4))
\/ (
LSeg (p00,p2)))) by
XBOOLE_1: 4
.= (((
LSeg (p00,p2))
\/ (
LSeg (p01,p2)))
\/ (L2
\/ (L3
\/ L4))) by
XBOOLE_1: 4
.=
R^2-unit_square by
A27,
TOPREAL1:def 2,
XBOOLE_1: 4;
A41:
{p1}
= ((
LSeg (p1,p01))
/\ (
LSeg (p1,p11))) by
A3,
TOPREAL1: 8;
A42: (P1
/\ P2)
= (((
LSeg (p1,p01))
/\ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p01))
/\ (
LSeg (p1,p11)))
\/ ((
LSeg (p1,p01))
/\ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ (L3
\/ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
A41,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p01,p2))
/\ ((
LSeg (p1,p11))
\/ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p01))
/\ L3)
\/ ((
LSeg (p1,p01))
/\ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
\/ ((
LSeg (p01,p2))
/\ ((L3
\/ L4)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ L4)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
\/ (((
LSeg (p01,p2))
/\ (L3
\/ L4))
\/
{p2}))) by
A12,
A28,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ L4)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
\/ ((((
LSeg (p01,p2))
/\ L3)
\/ ((
LSeg (p01,p2))
/\ L4))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p01))
/\ L4)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2}))) by
A40;
A43:
now
per cases ;
suppose
A44: p1
= p01;
then p1
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A45: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
<>
{} by
A44,
Lm23,
XBOOLE_0:def 4;
((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
c=
{p1} by
A39,
A44,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A46: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
=
{p1} by
A45,
ZFMISC_1: 33;
((
LSeg (p1,p01))
/\ L4)
= (
{p1}
/\ L4) by
A44,
RLTOPSP1: 70;
then ((
LSeg (p1,p01))
/\ L4)
=
{} by
A44,
Lm1,
Lm15;
hence (P1
/\ P2)
= ((
{p1}
\/ (
{p1}
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A42,
A46,
XBOOLE_1: 4
.= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2}));
end;
suppose
A47: p1
= p11;
then
A48: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
= ((
LSeg (p01,p2))
/\
{p1}) by
RLTOPSP1: 70;
not p1
in (
LSeg (p01,p2)) by
A31,
A47,
Lm6,
Lm10,
TOPREAL1: 3;
then ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
=
{} by
A48,
Lm1;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A42,
A47,
TOPREAL1: 18,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2}));
end;
suppose
A49: p1
<> p11 & p1
<> p01;
now
assume p01
in ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)));
then
A50: p01
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then (p1
`1 )
=
0 by
A13,
A15,
A50,
Lm6,
TOPREAL1: 3;
hence contradiction by
A13,
A16,
A49,
EUCLID: 53;
end;
then
A51:
{p01}
<> ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11))) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
c=
{p01} by
A8,
A39,
TOPREAL1: 15,
XBOOLE_1: 27;
then
A52: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p11)))
=
{} by
A51,
ZFMISC_1: 33;
now
assume p11
in ((
LSeg (p1,p01))
/\ L4);
then
A53: p11
in (
LSeg (p01,p1)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then (p11
`1 )
<= (p1
`1 ) by
A53,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A13,
A14,
Lm10,
XXREAL_0: 1;
hence contradiction by
A13,
A16,
A49,
EUCLID: 53;
end;
then
A54:
{p11}
<> ((
LSeg (p1,p01))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ L4)
c=
{p11} by
A11,
TOPREAL1: 18,
XBOOLE_1: 27;
then ((
LSeg (p1,p01))
/\ L4)
=
{} by
A54,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A42,
A52;
end;
end;
now
per cases ;
suppose
A55: p2
<> p00 & p2
<> p01;
now
assume p01
in ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)));
then
A56: p01
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p2
`2 ) by
A31,
EUCLID: 52;
then (p01
`2 )
<= (p2
`2 ) by
A56,
TOPREAL1: 4;
then (p2
`2 )
= 1 by
A31,
Lm7,
XXREAL_0: 1;
hence contradiction by
A31,
A55,
EUCLID: 53;
end;
then
A57:
{p01}
<> ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
c= (L2
/\ L1) by
A11,
A29,
XBOOLE_1: 27;
then
A58: ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
=
{} by
A57,
TOPREAL1: 15,
ZFMISC_1: 33;
now
assume p00
in ((
LSeg (p01,p2))
/\ L3);
then
A59: p00
in (
LSeg (p2,p01)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A31,
EUCLID: 52;
then
0
= (p2
`2 ) by
A31,
A59,
Lm5,
TOPREAL1: 4;
hence contradiction by
A31,
A55,
EUCLID: 53;
end;
then
A60:
{p00}
<> ((
LSeg (p01,p2))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ L3)
c=
{p00} by
A39,
TOPREAL1: 17,
XBOOLE_1: 27;
then ((
LSeg (p01,p2))
/\ L3)
=
{} by
A60,
ZFMISC_1: 33;
hence thesis by
A43,
A58,
ENUMSET1: 1;
end;
suppose
A61: p2
= p00;
then
A62: ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
= ((
LSeg (p1,p01))
/\
{p00}) by
RLTOPSP1: 70;
not p00
in (
LSeg (p1,p01)) by
A11,
Lm5,
Lm7,
Lm11,
TOPREAL1: 4;
then ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
=
{} by
A62,
Lm1;
hence thesis by
A43,
A61,
ENUMSET1: 1,
TOPREAL1: 17;
end;
suppose
A63: p2
= p01;
then p2
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
then
A64:
{}
<> ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2))) by
A63,
Lm22,
XBOOLE_0:def 4;
((
LSeg (p01,p2))
/\ L3)
= (
{p01}
/\ L3) by
A63,
RLTOPSP1: 70;
then
A65: ((
LSeg (p01,p2))
/\ L3)
=
{} by
Lm1,
Lm14;
((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
c= (L2
/\ L1) by
A11,
A29,
XBOOLE_1: 27;
then ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
=
{p2} by
A63,
A64,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A43,
A65,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A66: p2
in L2;
A67: q1
=
|[(q1
`1 ), (q1
`2 )]| by
EUCLID: 53;
A68: (
LSeg (p1,p2))
c= L2 by
A3,
A66,
TOPREAL1: 6;
consider q such that
A69: q
= p2 and
A70: (q
`1 )
<= 1 and
A71: (q
`1 )
>=
0 and
A72: (q
`2 )
= 1 by
A66,
TOPREAL1: 13;
A73: q
=
|[(q
`1 ), (q
`2 )]| by
EUCLID: 53;
now
per cases by
A1,
A13,
A16,
A69,
A72,
A67,
A73,
XXREAL_0: 1;
suppose
A74: (q1
`1 )
< (q
`1 );
A75: ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)))
c=
{p2}
proof
let a be
object;
assume
A76: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A77: p
in (
LSeg (p2,p11)) by
A76,
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A69,
A70,
EUCLID: 52;
then
A78: (p2
`1 )
<= (p
`1 ) by
A77,
TOPREAL1: 3;
A79: p
in (
LSeg (p1,p2)) by
A76,
XBOOLE_0:def 4;
then
A80: (p
`2 )
<= (p2
`2 ) by
A13,
A16,
A69,
A72,
TOPREAL1: 4;
(p
`1 )
<= (p2
`1 ) by
A13,
A69,
A74,
A79,
TOPREAL1: 3;
then
A81: (p2
`1 )
= (p
`1 ) by
A78,
XXREAL_0: 1;
(p1
`2 )
<= (p
`2 ) by
A13,
A16,
A69,
A72,
A79,
TOPREAL1: 4;
then (p
`2 )
= 1 by
A13,
A16,
A69,
A72,
A80,
XXREAL_0: 1;
then p
=
|[(p2
`1 ), 1]| by
A81,
EUCLID: 53
.= p2 by
A69,
A72,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A82: ((
LSeg (p1,p01))
/\ L3)
=
{} by
A11,
XBOOLE_1: 3,
XBOOLE_1: 26;
A83:
now
set a = the
Element of ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)));
assume
A84: ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A85: p
in (
LSeg (p01,p1)) by
A84,
XBOOLE_0:def 4;
A86: p
in (
LSeg (p2,p11)) by
A84,
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A69,
A70,
EUCLID: 52;
then
A87: (p2
`1 )
<= (p
`1 ) by
A86,
TOPREAL1: 3;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then (p
`1 )
<= (p1
`1 ) by
A85,
TOPREAL1: 3;
hence contradiction by
A13,
A69,
A74,
A87,
XXREAL_0: 2;
end;
A88: ((L1
\/ L3)
/\ L4)
= ((L1
/\ L4)
\/ (L3
/\ L4)) by
XBOOLE_1: 23
.=
{p10} by
Lm3,
TOPREAL1: 16;
(L1
\/ L3)
is_an_arc_of (p01,p10) by
Lm5,
Lm7,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 17;
then
A89: ((L1
\/ L3)
\/ L4)
is_an_arc_of (p01,p11) by
A88,
TOPREAL1: 10;
A90: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p01)))
c=
{p1}
proof
let a be
object;
assume
A91: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p01)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A92: p
in (
LSeg (p01,p1)) by
A91,
XBOOLE_0:def 4;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then
A93: (p
`1 )
<= (p1
`1 ) by
A92,
TOPREAL1: 3;
A94: p
in (
LSeg (p1,p2)) by
A91,
XBOOLE_0:def 4;
then
A95: (p
`2 )
<= (p2
`2 ) by
A13,
A16,
A69,
A72,
TOPREAL1: 4;
(p1
`1 )
<= (p
`1 ) by
A13,
A69,
A74,
A94,
TOPREAL1: 3;
then
A96: (p1
`1 )
= (p
`1 ) by
A93,
XXREAL_0: 1;
(p1
`2 )
<= (p
`2 ) by
A13,
A16,
A69,
A72,
A94,
TOPREAL1: 4;
then (p
`2 )
= 1 by
A13,
A16,
A69,
A72,
A95,
XXREAL_0: 1;
then p
=
|[(p1
`1 ), 1]| by
A96,
EUCLID: 53
.= p1 by
A13,
A16,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A97: ((
LSeg (p1,p2))
/\ L1)
c= (L2
/\ L1) by
A3,
A66,
TOPREAL1: 6,
XBOOLE_1: 26;
now
assume p11
in ((
LSeg (p1,p01))
/\ L4);
then
A98: p11
in (
LSeg (p01,p1)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then (p11
`1 )
<= (p1
`1 ) by
A98,
TOPREAL1: 3;
hence contradiction by
A13,
A14,
A70,
A74,
Lm10,
XXREAL_0: 1;
end;
then
A99:
{p11}
<> ((
LSeg (p1,p01))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p01))
/\ L4)
c=
{p11} by
A3,
Lm23,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
then
A100: ((
LSeg (p1,p01))
/\ L4)
=
{} by
A99,
ZFMISC_1: 33;
p01
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
then
A101: ((
LSeg (p1,p01))
/\ L1)
<>
{} by
Lm22,
XBOOLE_0:def 4;
now
assume p01
in (L1
/\ (
LSeg (p11,p2)));
then
A102: p01
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A69,
A70,
EUCLID: 52;
hence contradiction by
A15,
A69,
A74,
A102,
Lm6,
TOPREAL1: 3;
end;
then
A103:
{p01}
<> (L1
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
(L1
/\ (
LSeg (p11,p2)))
c=
{p01} by
A66,
Lm26,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
then
A104: (L1
/\ (
LSeg (p11,p2)))
=
{} by
A103,
ZFMISC_1: 33;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p01))
\/ (((L1
\/ L3)
\/ L4)
\/ (
LSeg (p11,p2))));
A105: p1
in (
LSeg (p1,p01)) by
RLTOPSP1: 68;
A106: ((
LSeg (p1,p01))
/\ L1)
c= (L2
/\ L1) by
A3,
Lm23,
TOPREAL1: 6,
XBOOLE_1: 26;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A107: (L4
/\ (
LSeg (p11,p2)))
<>
{} by
Lm27,
XBOOLE_0:def 4;
(L4
/\ (
LSeg (p11,p2)))
c= (L4
/\ L2) by
A66,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A108: (L4
/\ (
LSeg (p11,p2)))
=
{p11} by
A107,
TOPREAL1: 18,
ZFMISC_1: 33;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A109: (L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
(L3
/\ (
LSeg (p11,p2)))
c= (L3
/\ L2) by
A66,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A110: (L3
/\ (
LSeg (p11,p2)))
=
{} by
A109,
XBOOLE_1: 3;
(((L1
\/ L3)
\/ L4)
/\ (
LSeg (p11,p2)))
= (((L1
\/ L3)
/\ (
LSeg (p11,p2)))
\/ (L4
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((L1
/\ (
LSeg (p11,p2)))
\/ (L3
/\ (
LSeg (p11,p2))))
\/
{p11}) by
A108,
XBOOLE_1: 23
.=
{p11} by
A104,
A110;
then
A111: (((L1
\/ L3)
\/ L4)
\/ (
LSeg (p11,p2)))
is_an_arc_of (p01,p2) by
A89,
TOPREAL1: 10;
((
LSeg (p1,p01))
/\ (((L1
\/ L3)
\/ L4)
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p01))
/\ ((L1
\/ L3)
\/ L4))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p01))
/\ (L1
\/ L3))
\/ ((
LSeg (p1,p01))
/\ L4)) by
A83,
XBOOLE_1: 23
.= (((
LSeg (p1,p01))
/\ L1)
\/ ((
LSeg (p1,p01))
/\ L3)) by
A100,
XBOOLE_1: 23
.=
{p01} by
A82,
A106,
A101,
TOPREAL1: 15,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A111,
TOPREAL1: 11;
thus (P1
\/ P2)
= (((
LSeg (p01,p1))
\/ (
LSeg (p1,p2)))
\/ (((L1
\/ L3)
\/ L4)
\/ (
LSeg (p11,p2)))) by
XBOOLE_1: 4
.= ((((
LSeg (p01,p1))
\/ (
LSeg (p1,p2)))
\/ (
LSeg (p2,p11)))
\/ ((L1
\/ L3)
\/ L4)) by
XBOOLE_1: 4
.= (L2
\/ ((L1
\/ L3)
\/ L4)) by
A3,
A66,
TOPREAL1: 7
.= (L2
\/ (L1
\/ (L3
\/ L4))) by
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A112: p2
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2))) by
A112,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
then
A113: ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)))
=
{p2} by
A75,
XBOOLE_0:def 10;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A114: ((
LSeg (p1,p2))
/\ L3)
=
{} by
A68,
XBOOLE_1: 3,
XBOOLE_1: 26;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p01))) by
A105,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
then
A115: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p01)))
=
{p1} by
A90,
XBOOLE_0:def 10;
A116: (P1
/\ P2)
= (((
LSeg (p1,p2))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p1,p2))
/\ (((L1
\/ L3)
\/ L4)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L1
\/ L3)
\/ L4))
\/
{p2})) by
A115,
A113,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L1
\/ L3))
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L1)
\/ ((
LSeg (p1,p2))
/\ L3))
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L1)
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2}))) by
A114,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L1))
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
XBOOLE_1: 4;
A117:
now
per cases ;
suppose
A118: p1
= p01;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L1)
<>
{} by
A118,
Lm22,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L1)
=
{p1} by
A97,
A118,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
A116;
end;
suppose
A119: p1
<> p01;
now
assume p01
in ((
LSeg (p1,p2))
/\ L1);
then p01
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p1
`1 )
=
0 by
A13,
A15,
A69,
A74,
Lm6,
TOPREAL1: 3;
hence contradiction by
A13,
A16,
A119,
EUCLID: 53;
end;
then
{p01}
<> ((
LSeg (p1,p2))
/\ L1) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L1)
=
{} by
A97,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
A116;
end;
end;
A120: ((
LSeg (p1,p2))
/\ L4)
c=
{p11} by
A3,
A66,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
now
per cases ;
suppose
A121: p2
= p11;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L4)
<>
{} by
A121,
Lm27,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L4)
=
{p2} by
A120,
A121,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A117,
ENUMSET1: 1;
end;
suppose
A122: p2
<> p11;
now
assume p11
in ((
LSeg (p1,p2))
/\ L4);
then p11
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p11
`1 )
<= (p2
`1 ) by
A13,
A69,
A74,
TOPREAL1: 3;
then (p2
`1 )
= 1 by
A69,
A70,
Lm10,
XXREAL_0: 1;
hence contradiction by
A69,
A72,
A122,
EUCLID: 53;
end;
then
{p11}
<> ((
LSeg (p1,p2))
/\ L4) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L4)
=
{} by
A120,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A117,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
suppose
A123: (q
`1 )
< (q1
`1 );
A124: ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)))
c=
{p2}
proof
let a be
object;
assume
A125: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A126: p
in (
LSeg (p01,p2)) by
A125,
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A69,
A71,
EUCLID: 52;
then
A127: (p
`1 )
<= (p2
`1 ) by
A126,
TOPREAL1: 3;
A128: p
in (
LSeg (p2,p1)) by
A125,
XBOOLE_0:def 4;
then
A129: (p
`2 )
<= (p1
`2 ) by
A13,
A16,
A69,
A72,
TOPREAL1: 4;
(p2
`1 )
<= (p
`1 ) by
A13,
A69,
A123,
A128,
TOPREAL1: 3;
then
A130: (p2
`1 )
= (p
`1 ) by
A127,
XXREAL_0: 1;
(p2
`2 )
<= (p
`2 ) by
A13,
A16,
A69,
A72,
A128,
TOPREAL1: 4;
then (p
`2 )
= 1 by
A13,
A16,
A69,
A72,
A129,
XXREAL_0: 1;
then p
=
|[(p2
`1 ), 1]| by
A130,
EUCLID: 53
.= p2 by
A69,
A72,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A131: ((
LSeg (p1,p11))
/\ L3)
=
{} by
A8,
XBOOLE_1: 3,
XBOOLE_1: 26;
A132:
now
set a = the
Element of ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)));
assume
A133: ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A134: p
in (
LSeg (p1,p11)) by
A133,
XBOOLE_0:def 4;
A135: p
in (
LSeg (p01,p2)) by
A133,
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A69,
A71,
EUCLID: 52;
then
A136: (p
`1 )
<= (p2
`1 ) by
A135,
TOPREAL1: 3;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then (p1
`1 )
<= (p
`1 ) by
A134,
TOPREAL1: 3;
hence contradiction by
A13,
A69,
A123,
A136,
XXREAL_0: 2;
end;
A137: ((L4
\/ L3)
/\ L1)
= ((L1
/\ L4)
\/ (L3
/\ L1)) by
XBOOLE_1: 23
.=
{p00} by
Lm3,
TOPREAL1: 17;
(L4
\/ L3)
is_an_arc_of (p11,p00) by
Lm9,
Lm11,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 16;
then
A138: ((L4
\/ L3)
\/ L1)
is_an_arc_of (p11,p01) by
A137,
TOPREAL1: 10;
now
assume p11
in (L4
/\ (
LSeg (p01,p2)));
then
A139: p11
in (
LSeg (p01,p2)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A69,
A71,
EUCLID: 52;
then (p11
`1 )
<= (p2
`1 ) by
A139,
TOPREAL1: 3;
hence contradiction by
A14,
A69,
A70,
A123,
Lm10,
XXREAL_0: 1;
end;
then
A140:
{p11}
<> (L4
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
(L4
/\ (
LSeg (p01,p2)))
c= (L4
/\ L2) by
A66,
Lm23,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A141: (L4
/\ (
LSeg (p01,p2)))
=
{} by
A140,
TOPREAL1: 18,
ZFMISC_1: 33;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A142: ((
LSeg (p1,p11))
/\ L4)
<>
{} by
Lm27,
XBOOLE_0:def 4;
now
assume p01
in ((
LSeg (p1,p11))
/\ L1);
then
A143: p01
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
hence contradiction by
A13,
A71,
A123,
A143,
Lm6,
TOPREAL1: 3;
end;
then
A144:
{p01}
<> ((
LSeg (p1,p11))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L1)
c= (L2
/\ L1) by
A3,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A145: ((
LSeg (p1,p11))
/\ L1)
=
{} by
A144,
TOPREAL1: 15,
ZFMISC_1: 33;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A146: ((
LSeg (p1,p2))
/\ L3)
=
{} by
A68,
XBOOLE_1: 3,
XBOOLE_1: 26;
A147: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)))
c=
{p1}
proof
let a be
object;
assume
A148: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A149: p
in (
LSeg (p1,p11)) by
A148,
XBOOLE_0:def 4;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then
A150: (p1
`1 )
<= (p
`1 ) by
A149,
TOPREAL1: 3;
A151: p
in (
LSeg (p2,p1)) by
A148,
XBOOLE_0:def 4;
then
A152: (p
`2 )
<= (p1
`2 ) by
A13,
A16,
A69,
A72,
TOPREAL1: 4;
(p
`1 )
<= (p1
`1 ) by
A13,
A69,
A123,
A151,
TOPREAL1: 3;
then
A153: (p1
`1 )
= (p
`1 ) by
A150,
XXREAL_0: 1;
(p2
`2 )
<= (p
`2 ) by
A13,
A16,
A69,
A72,
A151,
TOPREAL1: 4;
then (p
`2 )
= 1 by
A13,
A16,
A69,
A72,
A152,
XXREAL_0: 1;
then p
=
|[(p1
`1 ), 1]| by
A153,
EUCLID: 53
.= p1 by
A13,
A16,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A154: ((
LSeg (p1,p11))
/\ L4)
c=
{p11} by
A3,
Lm26,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A155: (L1
/\ (
LSeg (p01,p2)))
<>
{} by
Lm22,
XBOOLE_0:def 4;
(L1
/\ (
LSeg (p01,p2)))
c=
{p01} by
A66,
Lm23,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
then
A156: (L1
/\ (
LSeg (p01,p2)))
=
{p01} by
A155,
ZFMISC_1: 33;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p11))
\/ (((L4
\/ L3)
\/ L1)
\/ (
LSeg (p01,p2))));
A157: p1
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A158: (L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
(L3
/\ (
LSeg (p01,p2)))
c= (L3
/\ L2) by
A66,
Lm23,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A159: (L3
/\ (
LSeg (p01,p2)))
=
{} by
A158,
XBOOLE_1: 3;
(((L4
\/ L3)
\/ L1)
/\ (
LSeg (p01,p2)))
= (((L4
\/ L3)
/\ (
LSeg (p01,p2)))
\/ (L1
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= (((L4
/\ (
LSeg (p01,p2)))
\/ (L3
/\ (
LSeg (p01,p2))))
\/
{p01}) by
A156,
XBOOLE_1: 23
.=
{p01} by
A141,
A159;
then
A160: (((L4
\/ L3)
\/ L1)
\/ (
LSeg (p01,p2)))
is_an_arc_of (p11,p2) by
A138,
TOPREAL1: 10;
((
LSeg (p1,p11))
/\ (((L4
\/ L3)
\/ L1)
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p11))
/\ ((L4
\/ L3)
\/ L1))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ (L4
\/ L3))
\/ ((
LSeg (p1,p11))
/\ L1)) by
A132,
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ L4)
\/ ((
LSeg (p1,p11))
/\ L3)) by
A145,
XBOOLE_1: 23
.=
{p11} by
A131,
A154,
A142,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A160,
TOPREAL1: 11;
thus (P1
\/ P2)
= (((
LSeg (p2,p1))
\/ (
LSeg (p1,p11)))
\/ (((L4
\/ L3)
\/ L1)
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= (((
LSeg (p01,p2))
\/ ((
LSeg (p2,p1))
\/ (
LSeg (p1,p11))))
\/ ((L4
\/ L3)
\/ L1)) by
XBOOLE_1: 4
.= (L2
\/ ((L4
\/ L3)
\/ L1)) by
A3,
A66,
TOPREAL1: 7
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A161: p2
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))) by
A161,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
then
A162: ((
LSeg (p1,p2))
/\ (
LSeg (p01,p2)))
=
{p2} by
A124,
XBOOLE_0:def 10;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11))) by
A157,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11))) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)))
=
{p1} by
A147,
XBOOLE_0:def 10;
then
A163: (P1
/\ P2)
= (
{p1}
\/ ((
LSeg (p1,p2))
/\ (((L4
\/ L3)
\/ L1)
\/ (
LSeg (p01,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L4
\/ L3)
\/ L1))
\/
{p2})) by
A162,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L4
\/ L3))
\/ ((
LSeg (p1,p2))
/\ L1))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L4)
\/ ((
LSeg (p1,p2))
/\ L3))
\/ ((
LSeg (p1,p2))
/\ L1))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/ (((
LSeg (p1,p2))
/\ L1)
\/
{p2}))) by
A146,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/ (((
LSeg (p1,p2))
/\ L1)
\/
{p2})) by
XBOOLE_1: 4;
A164: ((
LSeg (p1,p2))
/\ L1)
c= (L2
/\ L1) by
A3,
A66,
TOPREAL1: 6,
XBOOLE_1: 26;
A165:
now
per cases ;
suppose
A166: p2
= p01;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L1)
<>
{} by
A166,
Lm22,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L1)
=
{p2} by
A164,
A166,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2}) by
A163;
end;
suppose
A167: p2
<> p01;
now
assume p01
in ((
LSeg (p1,p2))
/\ L1);
then p01
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p2
`1 )
=
0 by
A13,
A69,
A71,
A123,
Lm6,
TOPREAL1: 3;
hence contradiction by
A69,
A72,
A167,
EUCLID: 53;
end;
then
{p01}
<> ((
LSeg (p1,p2))
/\ L1) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L1)
=
{} by
A164,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2}) by
A163;
end;
end;
A168: ((
LSeg (p1,p2))
/\ L4)
c=
{p11} by
A3,
A66,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
now
per cases ;
suppose
A169: p1
= p11;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L4)
<>
{} by
A169,
Lm27,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L4)
=
{p1} by
A168,
A169,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A165,
ENUMSET1: 1;
end;
suppose
A170: p1
<> p11;
now
assume p11
in ((
LSeg (p1,p2))
/\ L4);
then p11
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p11
`1 )
<= (p1
`1 ) by
A13,
A69,
A123,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A13,
A14,
Lm10,
XXREAL_0: 1;
hence contradiction by
A13,
A16,
A170,
EUCLID: 53;
end;
then
{p11}
<> ((
LSeg (p1,p2))
/\ L4) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L4)
=
{} by
A168,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A165,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
end;
hence thesis;
end;
suppose
A171: p2
in L3;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A172: ((
LSeg (p01,p00))
/\ (
LSeg (p00,p2)))
<>
{} by
Lm20,
XBOOLE_0:def 4;
(
LSeg (p00,p2))
c= L3 by
A171,
Lm21,
TOPREAL1: 6;
then ((
LSeg (p01,p00))
/\ (
LSeg (p00,p2)))
c=
{p00} by
TOPREAL1: 17,
XBOOLE_1: 27;
then ((
LSeg (p01,p00))
/\ (
LSeg (p00,p2)))
=
{p00} by
A172,
ZFMISC_1: 33;
then
A173: (L1
\/ (
LSeg (p00,p2)))
is_an_arc_of (p01,p2) by
Lm5,
Lm7,
TOPREAL1: 12;
(
LSeg (p2,p00))
c= L3 by
A171,
Lm21,
TOPREAL1: 6;
then
A174: ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))
=
{} by
A11,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A175: p10
in ((
LSeg (p11,p10))
/\ (
LSeg (p10,p2))) by
Lm25,
XBOOLE_0:def 4;
(
LSeg (p10,p2))
c= L3 by
A171,
Lm24,
TOPREAL1: 6;
then ((
LSeg (p11,p10))
/\ (
LSeg (p10,p2)))
c= (L4
/\ L3) by
XBOOLE_1: 27;
then ((
LSeg (p11,p10))
/\ (
LSeg (p10,p2)))
=
{p10} by
A175,
TOPREAL1: 16,
ZFMISC_1: 33;
then
A176: (L4
\/ (
LSeg (p10,p2)))
is_an_arc_of (p11,p2) by
Lm9,
Lm11,
TOPREAL1: 12;
take P1 = (((
LSeg (p1,p11))
\/ L4)
\/ (
LSeg (p10,p2))), P2 = (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2)));
A177: ((
LSeg (p1,p11))
\/ (
LSeg (p1,p01)))
= L2 by
A3,
TOPREAL1: 5;
A178: (
LSeg (p2,p10))
c= L3 by
A171,
Lm24,
TOPREAL1: 6;
then
A179: ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
=
{} by
A8,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A180: (L2
/\ (
LSeg (p00,p2)))
c= (L2
/\ L3) by
A171,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
c= (L2
/\ (
LSeg (p00,p2))) by
A3,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A181: ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
=
{} by
A180,
Lm2,
XBOOLE_1: 1,
XBOOLE_1: 3;
A182: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A11,
A178,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A183: ((
LSeg (p10,p2))
/\ (
LSeg (p00,p2)))
=
{p2} by
A171,
TOPREAL1: 8;
((
LSeg (p1,p11))
/\ (L4
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p11))
/\ L4)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.=
{p11} by
A6,
A10,
A179,
ZFMISC_1: 33;
then ((
LSeg (p1,p11))
\/ (L4
\/ (
LSeg (p10,p2))))
is_an_arc_of (p1,p2) by
A176,
TOPREAL1: 11;
hence P1
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
A184: ex q2 st q2
= p2 & (q2
`1 )
<= 1 & (q2
`1 )
>=
0 & (q2
`2 )
=
0 by
A171,
TOPREAL1: 13;
((
LSeg (p1,p01))
/\ (L1
\/ (
LSeg (p00,p2))))
= (((
LSeg (p01,p1))
/\ L1)
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.=
{p01} by
A7,
A5,
A174,
TOPREAL1: 15,
ZFMISC_1: 33;
then ((
LSeg (p1,p01))
\/ (L1
\/ (
LSeg (p00,p2))))
is_an_arc_of (p1,p2) by
A173,
TOPREAL1: 11;
hence P2
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
((
LSeg (p10,p2))
\/ (
LSeg (p00,p2)))
= L3 by
A171,
TOPREAL1: 5;
hence
R^2-unit_square
= (L2
\/ ((L4
\/ ((
LSeg (p10,p2))
\/ (
LSeg (p00,p2))))
\/ L1)) by
TOPREAL1:def 2,
XBOOLE_1: 4
.= (L2
\/ (((L4
\/ (
LSeg (p10,p2)))
\/ (
LSeg (p00,p2)))
\/ L1)) by
XBOOLE_1: 4
.= (L2
\/ ((L4
\/ (
LSeg (p10,p2)))
\/ (L1
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 4
.= ((
LSeg (p1,p11))
\/ (((L4
\/ (
LSeg (p10,p2)))
\/ (L1
\/ (
LSeg (p00,p2))))
\/ (
LSeg (p1,p01)))) by
A177,
XBOOLE_1: 4
.= ((
LSeg (p1,p11))
\/ ((L4
\/ (
LSeg (p10,p2)))
\/ ((L1
\/ (
LSeg (p00,p2)))
\/ (
LSeg (p1,p01))))) by
XBOOLE_1: 4
.= (((
LSeg (p1,p11))
\/ (L4
\/ (
LSeg (p10,p2))))
\/ ((L1
\/ (
LSeg (p00,p2)))
\/ (
LSeg (p1,p01)))) by
XBOOLE_1: 4
.= ((((
LSeg (p1,p11))
\/ L4)
\/ (
LSeg (p10,p2)))
\/ ((
LSeg (p1,p01))
\/ (L1
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A185: ((
LSeg (p1,p11))
/\ (
LSeg (p1,p01)))
=
{p1} by
A3,
TOPREAL1: 8;
A186: (P1
/\ P2)
= ((((
LSeg (p1,p11))
\/ L4)
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))
\/ (L4
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p11))
/\ ((
LSeg (p1,p01))
\/ L1))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2))))
\/ (L4
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p11))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (L4
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
A181,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((L4
/\ ((
LSeg (p1,p01))
\/ L1))
\/ (L4
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
A185,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (((L4
/\ (
LSeg (p1,p01)))
\/ (L1
/\ L4))
\/ (L4
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ (((
LSeg (p1,p01))
\/ L1)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((L4
/\ (
LSeg (p1,p01)))
\/ (L4
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ ((
LSeg (p1,p01))
\/ L1))
\/
{p2})) by
A183,
Lm3,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((L4
/\ (
LSeg (p1,p01)))
\/ (L4
/\ (
LSeg (p00,p2)))))
\/ ((((
LSeg (p10,p2))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p10,p2))
/\ L1))
\/
{p2})) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((L4
/\ (
LSeg (p1,p01)))
\/ (L4
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A182;
A187:
now
per cases ;
suppose
A188: p1
= p01;
then (L4
/\ (
LSeg (p1,p01)))
= (L4
/\
{p01}) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm15;
hence (P1
/\ P2)
= ((
{p1}
\/ (L4
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A186,
A188,
TOPREAL1: 15;
end;
suppose
A189: p1
= p11;
then ((
LSeg (p1,p11))
/\ L1)
= (
{p11}
/\ L1) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm18;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ (L4
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A186,
A189,
TOPREAL1: 18,
XBOOLE_1: 4
.= ((
{p1}
\/ (L4
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2}));
end;
suppose
A190: p1
<> p11 & p1
<> p01;
now
assume p11
in (L4
/\ (
LSeg (p1,p01)));
then
A191: p11
in (
LSeg (p01,p1)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then 1
<= (p1
`1 ) by
A191,
Lm10,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A13,
A14,
XXREAL_0: 1;
hence contradiction by
A13,
A16,
A190,
EUCLID: 53;
end;
then
A192:
{p11}
<> (L4
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
(L4
/\ (
LSeg (p1,p01)))
c= (L4
/\ L2) by
A3,
Lm23,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A193: (L4
/\ (
LSeg (p1,p01)))
=
{} by
A192,
TOPREAL1: 18,
ZFMISC_1: 33;
now
assume p01
in ((
LSeg (p1,p11))
/\ L1);
then
A194: p01
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then (p1
`1 )
=
0 by
A13,
A15,
A194,
Lm6,
TOPREAL1: 3;
hence contradiction by
A13,
A16,
A190,
EUCLID: 53;
end;
then
A195:
{p01}
<> ((
LSeg (p1,p11))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L1)
c= (L2
/\ L1) by
A3,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p11))
/\ L1)
=
{} by
A195,
TOPREAL1: 15,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ (L4
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A186,
A193;
end;
end;
now
per cases ;
suppose
A196: p2
= p00;
then (L4
/\ (
LSeg (p00,p2)))
= (L4
/\
{p00}) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm12;
hence thesis by
A187,
A196,
ENUMSET1: 1,
TOPREAL1: 17;
end;
suppose
A197: p2
= p10;
then ((
LSeg (p10,p2))
/\ L1)
= (
{p10}
/\ L1) by
RLTOPSP1: 70
.=
{} by
Lm1,
Lm16;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A187,
A197,
TOPREAL1: 16,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A198: p2
<> p10 & p2
<> p00;
now
assume p00
in ((
LSeg (p10,p2))
/\ L1);
then
A199: p00
in (
LSeg (p2,p10)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A184,
EUCLID: 52;
then (p2
`1 )
=
0 by
A184,
A199,
Lm4,
TOPREAL1: 3;
hence contradiction by
A184,
A198,
EUCLID: 53;
end;
then
A200:
{p00}
<> ((
LSeg (p10,p2))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ L1)
c= (L3
/\ L1) by
A171,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A201: ((
LSeg (p10,p2))
/\ L1)
=
{} by
A200,
TOPREAL1: 17,
ZFMISC_1: 33;
now
assume p10
in (L4
/\ (
LSeg (p00,p2)));
then
A202: p10
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A184,
EUCLID: 52;
then 1
<= (p2
`1 ) by
A202,
Lm8,
TOPREAL1: 3;
then (p2
`1 )
= 1 by
A184,
XXREAL_0: 1;
hence contradiction by
A184,
A198,
EUCLID: 53;
end;
then
A203:
{p10}
<> (L4
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
(L4
/\ (
LSeg (p00,p2)))
c= (L4
/\ L3) by
A171,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
then (L4
/\ (
LSeg (p00,p2)))
=
{} by
A203,
TOPREAL1: 16,
ZFMISC_1: 33;
hence thesis by
A187,
A201,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A204: p2
in L4;
then
A205: (
LSeg (p11,p2))
c= L4 by
Lm27,
TOPREAL1: 6;
(
LSeg (p1,p11))
c= L2 by
A3,
Lm26,
TOPREAL1: 6;
then
A206: ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2)))
c= (L2
/\ L4) by
A205,
XBOOLE_1: 27;
take P1 = ((
LSeg (p1,p11))
\/ (
LSeg (p11,p2))), P2 = ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2))));
A207: p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A208: (L3
/\ (
LSeg (p10,p2)))
<>
{} by
Lm24,
XBOOLE_0:def 4;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2)))
<>
{} by
A207,
XBOOLE_0:def 4;
then
A209: ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2)))
=
{p11} by
A206,
TOPREAL1: 18,
ZFMISC_1: 33;
p1
<> p11 or p11
<> p2 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A209,
TOPREAL1: 12;
A210: L2
= ((
LSeg (p1,p11))
\/ (
LSeg (p1,p01))) by
A3,
TOPREAL1: 5;
A211: L3
is_an_arc_of (p00,p10) by
Lm4,
Lm8,
TOPREAL1: 9;
L1
is_an_arc_of (p01,p00) by
Lm5,
Lm7,
TOPREAL1: 9;
then
A212: (L1
\/ L3)
is_an_arc_of (p01,p10) by
A211,
TOPREAL1: 2,
TOPREAL1: 17;
A213: ((
LSeg (p11,p2))
/\ (
LSeg (p10,p2)))
=
{p2} by
A204,
TOPREAL1: 8;
A214: L4
= ((
LSeg (p10,p2))
\/ (
LSeg (p11,p2))) by
A204,
TOPREAL1: 5;
A215: (
LSeg (p10,p2))
c= L4 by
A204,
Lm25,
TOPREAL1: 6;
then
A216: (L3
/\ (
LSeg (p10,p2)))
c=
{p10} by
TOPREAL1: 16,
XBOOLE_1: 27;
A217: ex q st q
= p2 & (q
`1 )
= 1 & (q
`2 )
<= 1 & (q
`2 )
>=
0 by
A204,
TOPREAL1: 13;
now
A218: (p10
`2 )
<= (p2
`2 ) by
A217,
EUCLID: 52;
assume
A219: p11
in ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)));
then
A220: p11
in (
LSeg (p01,p1)) by
XBOOLE_0:def 4;
p11
in (
LSeg (p10,p2)) by
A219,
XBOOLE_0:def 4;
then (p11
`2 )
<= (p2
`2 ) by
A218,
TOPREAL1: 4;
then
A221: (p11
`2 )
= (p2
`2 ) by
A217,
Lm11,
XXREAL_0: 1;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then (p11
`1 )
<= (p1
`1 ) by
A220,
TOPREAL1: 3;
then (p11
`1 )
= (p1
`1 ) by
A13,
A14,
Lm10,
XXREAL_0: 1;
then p1
=
|[(p11
`1 ), (p11
`2 )]| by
A13,
A16,
Lm11,
EUCLID: 53
.= p2 by
A217,
A221,
Lm10,
EUCLID: 53;
hence contradiction by
A1;
end;
then
A222:
{p11}
<> ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
A223: (L1
/\ (
LSeg (p10,p2)))
=
{} by
A215,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L1
\/ L3)
/\ (
LSeg (p10,p2)))
= ((L1
/\ (
LSeg (p10,p2)))
\/ (L3
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.=
{p10} by
A223,
A216,
A208,
ZFMISC_1: 33;
then
A224: ((L1
\/ L3)
\/ (
LSeg (p10,p2)))
is_an_arc_of (p01,p2) by
A212,
TOPREAL1: 10;
A225: (
LSeg (p2,p11))
c= L4 by
A204,
Lm27,
TOPREAL1: 6;
then
A226: ((
LSeg (p11,p2))
/\ L1)
=
{} by
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))
c=
{p11} by
A11,
A215,
TOPREAL1: 18,
XBOOLE_1: 27;
then
A227: ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))
=
{} by
A222,
ZFMISC_1: 33;
((
LSeg (p1,p01))
/\ ((L1
\/ L3)
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p01))
/\ (L1
\/ L3))
\/ ((
LSeg (p1,p01))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p01))
/\ L1)
\/ ((
LSeg (p1,p01))
/\ L3)) by
A227,
XBOOLE_1: 23
.=
{p01} by
A12,
A7,
A5,
TOPREAL1: 15,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A224,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p11,p2))
\/ ((
LSeg (p1,p11))
\/ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 4
.= ((L2
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2))))
\/ (
LSeg (p11,p2))) by
A210,
XBOOLE_1: 4
.= (L2
\/ (((L1
\/ L3)
\/ (
LSeg (p10,p2)))
\/ (
LSeg (p11,p2)))) by
XBOOLE_1: 4
.= (L2
\/ ((L1
\/ L3)
\/ ((
LSeg (p10,p2))
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 4
.= (L2
\/ (L1
\/ (L3
\/ L4))) by
A214,
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A228:
{p1}
= ((
LSeg (p1,p11))
/\ (
LSeg (p1,p01))) by
A3,
TOPREAL1: 8;
A229: (P1
/\ P2)
= (((
LSeg (p1,p11))
/\ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p1,p11))
/\ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ (L1
\/ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
A228,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p11))
/\ L1)
\/ ((
LSeg (p1,p11))
/\ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p01))
\/ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p11))
/\ L1)
\/ ((
LSeg (p1,p11))
/\ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/ ((
LSeg (p11,p2))
/\ ((L1
\/ L3)
\/ (
LSeg (p10,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p11))
/\ L1)
\/ ((
LSeg (p1,p11))
/\ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/ (((
LSeg (p11,p2))
/\ (L1
\/ L3))
\/
{p2}))) by
A213,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p11))
/\ L1)
\/ ((
LSeg (p1,p11))
/\ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/ ((((
LSeg (p11,p2))
/\ L1)
\/ ((
LSeg (p11,p2))
/\ L3))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ L1)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/ (((
LSeg (p11,p2))
/\ L3)
\/
{p2}))) by
A9,
A226;
A230:
now
per cases ;
suppose
A231: p2
= p10;
then
A232: not p2
in (
LSeg (p1,p11)) by
A8,
Lm7,
Lm9,
Lm11,
TOPREAL1: 4;
((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
= ((
LSeg (p1,p11))
/\
{p2}) by
A231,
RLTOPSP1: 70
.=
{} by
A232,
Lm1;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A229,
A231,
TOPREAL1: 16;
end;
suppose
A233: p2
= p11;
then p2
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A234: ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
<>
{} by
A233,
Lm27,
XBOOLE_0:def 4;
((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
c=
{p2} by
A8,
A233,
TOPREAL1: 18,
XBOOLE_1: 27;
then
A235: ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
=
{p2} by
A234,
ZFMISC_1: 33;
((
LSeg (p11,p2))
/\ L3)
= (
{p11}
/\ L3) by
A233,
RLTOPSP1: 70
.=
{} by
Lm1,
Lm19;
hence (P1
/\ P2)
= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/
{p2})
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A229,
A235,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/ (
{p2}
\/
{p2}))) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/
{p2}));
end;
suppose
A236: p2
<> p11 & p2
<> p10;
now
assume p11
in ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)));
then
A237: p11
in (
LSeg (p10,p2)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A217,
EUCLID: 52;
then (p11
`2 )
<= (p2
`2 ) by
A237,
TOPREAL1: 4;
then (p2
`2 )
= 1 by
A217,
Lm11,
XXREAL_0: 1;
hence contradiction by
A217,
A236,
EUCLID: 53;
end;
then
A238:
{p11}
<> ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
c=
{p11} by
A8,
A215,
TOPREAL1: 18,
XBOOLE_1: 27;
then
A239: ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
=
{} by
A238,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p11,p2))
/\ L3);
then
A240: p10
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A217,
EUCLID: 52;
then (p2
`2 )
=
0 by
A217,
A240,
Lm9,
TOPREAL1: 4;
hence contradiction by
A217,
A236,
EUCLID: 53;
end;
then
A241:
{p10}
<> ((
LSeg (p11,p2))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p11,p2))
/\ L3)
c= (L4
/\ L3) by
A225,
XBOOLE_1: 27;
then ((
LSeg (p11,p2))
/\ L3)
=
{} by
A241,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ L1))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
\/
{p2})) by
A229,
A239;
end;
end;
now
per cases ;
suppose
A242: p1
= p01;
then
A243: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
= ((
LSeg (p11,p2))
/\
{p1}) by
RLTOPSP1: 70;
p1
in (
LSeg (p11,p2)) implies contradiction by
A225,
A242,
Lm6,
Lm8,
Lm10,
TOPREAL1: 3;
then ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A243,
Lm1;
hence thesis by
A230,
A242,
ENUMSET1: 1,
TOPREAL1: 15;
end;
suppose
A244: p1
= p11;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A245: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
<>
{} by
A244,
Lm26,
XBOOLE_0:def 4;
((
LSeg (p1,p11))
/\ L1)
= (
{p1}
/\ L1) by
A244,
RLTOPSP1: 70;
then
A246: ((
LSeg (p1,p11))
/\ L1)
=
{} by
A244,
Lm1,
Lm18;
((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
c= (L4
/\ L2) by
A11,
A225,
XBOOLE_1: 27;
then ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
=
{p1} by
A244,
A245,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/
{p1})
\/
{p2}) by
A230,
A246,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A247: p1
<> p11 & p1
<> p01;
now
assume p11
in ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)));
then
A248: p11
in (
LSeg (p01,p1)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p1
`1 ) by
A13,
A15,
EUCLID: 52;
then (p11
`1 )
<= (p1
`1 ) by
A248,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A13,
A14,
Lm10,
XXREAL_0: 1;
hence contradiction by
A13,
A16,
A247,
EUCLID: 53;
end;
then
A249:
{p11}
<> ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01))) by
ZFMISC_1: 31;
((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
c= (L4
/\ L2) by
A11,
A225,
XBOOLE_1: 27;
then
A250: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p01)))
=
{} by
A249,
TOPREAL1: 18,
ZFMISC_1: 33;
now
assume p01
in ((
LSeg (p1,p11))
/\ L1);
then
A251: p01
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p11
`1 ) by
A13,
A14,
EUCLID: 52;
then (p1
`1 )
=
0 by
A13,
A15,
A251,
Lm6,
TOPREAL1: 3;
hence contradiction by
A13,
A16,
A247,
EUCLID: 53;
end;
then
A252:
{p01}
<> ((
LSeg (p1,p11))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L1)
c= (L2
/\ L1) by
A8,
XBOOLE_1: 27;
then ((
LSeg (p1,p11))
/\ L1)
=
{} by
A252,
TOPREAL1: 15,
ZFMISC_1: 33;
hence thesis by
A230,
A250,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
end;
Lm32: p1
<> p2 & p2
in
R^2-unit_square & p1
in (
LSeg (p00,p10)) implies ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) &
R^2-unit_square
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume that
A1: p1
<> p2 and
A2: p2
in
R^2-unit_square and
A3: p1
in (
LSeg (p00,p10));
A4: p2
in (L1
\/ L2) or p2
in (L3
\/ L4) by
A2,
TOPREAL1:def 2,
XBOOLE_0:def 3;
A5: ((
LSeg (p10,p1))
/\ L4)
c= (L3
/\ L4) by
A3,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then
A6: p00
in ((
LSeg (p1,p00))
/\ L1) by
Lm20,
XBOOLE_0:def 4;
p10
in (
LSeg (p10,p1)) by
RLTOPSP1: 68;
then
A7: ((
LSeg (p10,p1))
/\ L4)
<>
{} by
Lm25,
XBOOLE_0:def 4;
A8: ((
LSeg (p1,p00))
/\ L1)
c= (L3
/\ L1) by
A3,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
A9: ((
LSeg (p1,p00))
/\ (
LSeg (p1,p10)))
=
{p1} by
A3,
TOPREAL1: 8;
A10: (
LSeg (p00,p1))
c= L3 by
A3,
Lm21,
TOPREAL1: 6;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A11: ((
LSeg (p1,p00))
/\ L2)
=
{} by
A10,
XBOOLE_1: 3,
XBOOLE_1: 26;
A12: (
LSeg (p10,p1))
c= L3 by
A3,
Lm24,
TOPREAL1: 6;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A13: ((
LSeg (p10,p1))
/\ L2)
=
{} by
A12,
XBOOLE_1: 3,
XBOOLE_1: 26;
consider p such that
A14: p
= p1 and
A15: (p
`1 )
<= 1 and
A16: (p
`1 )
>=
0 and
A17: (p
`2 )
=
0 by
A3,
TOPREAL1: 13;
per cases by
A4,
XBOOLE_0:def 3;
suppose
A18: p2
in L1;
A19: L2
is_an_arc_of (p11,p01) by
Lm6,
Lm10,
TOPREAL1: 9;
L4
is_an_arc_of (p10,p11) by
Lm9,
Lm11,
TOPREAL1: 9;
then
A20: (L4
\/ L2)
is_an_arc_of (p10,p01) by
A19,
TOPREAL1: 2,
TOPREAL1: 18;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A21: ((
LSeg (p1,p00))
/\ L2)
=
{} by
A10,
XBOOLE_1: 3,
XBOOLE_1: 26;
take P1 = ((
LSeg (p1,p00))
\/ (
LSeg (p00,p2))), P2 = ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2))));
A22: ((
LSeg (p1,p00))
\/ (
LSeg (p1,p10)))
= L3 by
A3,
TOPREAL1: 5;
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A23: p01
in (L2
/\ (
LSeg (p01,p2))) by
Lm23,
XBOOLE_0:def 4;
A24: p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then
A25: p00
in ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2))) by
A24,
XBOOLE_0:def 4;
A26: (
LSeg (p00,p2))
c= L1 by
A18,
Lm20,
TOPREAL1: 6;
then ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2)))
c= (L3
/\ L1) by
A10,
XBOOLE_1: 27;
then
A27: ((
LSeg (p1,p00))
/\ (
LSeg (p00,p2)))
=
{p00} by
A25,
TOPREAL1: 17,
ZFMISC_1: 33;
A28: ex q st q
= p2 & (q
`1 )
=
0 & (q
`2 )
<= 1 & (q
`2 )
>=
0 by
A18,
TOPREAL1: 13;
now
A29: (p2
`2 )
<= (p01
`2 ) by
A28,
EUCLID: 52;
assume
A30: p00
in ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)));
then
A31: p00
in (
LSeg (p1,p10)) by
XBOOLE_0:def 4;
p00
in (
LSeg (p2,p01)) by
A30,
XBOOLE_0:def 4;
then
A32:
0
= (p2
`2 ) by
A28,
A29,
Lm5,
TOPREAL1: 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then
0
= (p1
`1 ) by
A14,
A16,
A31,
Lm4,
TOPREAL1: 3;
then p1
= p00 by
A14,
A17,
EUCLID: 53
.= p2 by
A28,
A32,
EUCLID: 53;
hence contradiction by
A1;
end;
then
A33:
{p00}
<> ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
p1
<> p00 or p00
<> p2 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A27,
TOPREAL1: 12;
A34:
{p1}
= ((
LSeg (p1,p00))
/\ (
LSeg (p1,p10))) by
A3,
TOPREAL1: 8;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A35: ((
LSeg (p00,p2))
/\ L4)
=
{} by
A26,
XBOOLE_1: 3,
XBOOLE_1: 26;
A36: (
LSeg (p2,p01))
c= L1 by
A18,
Lm22,
TOPREAL1: 6;
then
A37: (L2
/\ (
LSeg (p01,p2)))
c= (L2
/\ L1) by
XBOOLE_1: 27;
A38: (L4
/\ (
LSeg (p01,p2)))
=
{} by
A36,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L4
\/ L2)
/\ (
LSeg (p01,p2)))
= ((L4
/\ (
LSeg (p01,p2)))
\/ (L2
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.=
{p01} by
A38,
A37,
A23,
TOPREAL1: 15,
ZFMISC_1: 33;
then
A39: ((L4
\/ L2)
\/ (
LSeg (p01,p2)))
is_an_arc_of (p10,p2) by
A20,
TOPREAL1: 10;
A40:
{p2}
= ((
LSeg (p00,p2))
/\ (
LSeg (p01,p2))) by
A18,
TOPREAL1: 8;
A41: ((
LSeg (p01,p2))
\/ (
LSeg (p00,p2)))
= L1 by
A18,
TOPREAL1: 5;
((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))
c= (L3
/\ L1) by
A12,
A36,
XBOOLE_1: 27;
then
A42: ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))
=
{} by
A33,
TOPREAL1: 17,
ZFMISC_1: 33;
((
LSeg (p1,p10))
/\ ((L4
\/ L2)
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p10))
/\ (L4
\/ L2))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ L4)
\/ ((
LSeg (p10,p1))
/\ L2)) by
A42,
XBOOLE_1: 23
.=
{p10} by
A13,
A5,
A7,
TOPREAL1: 16,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A39,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p00,p2))
\/ ((
LSeg (p1,p00))
\/ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 4
.= ((
LSeg (p00,p2))
\/ (L3
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2))))) by
A22,
XBOOLE_1: 4
.= ((
LSeg (p00,p2))
\/ ((L3
\/ (L4
\/ L2))
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= ((
LSeg (p00,p2))
\/ (((L3
\/ L4)
\/ L2)
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= ((
LSeg (p00,p2))
\/ ((L3
\/ L4)
\/ (L2
\/ (
LSeg (p01,p2))))) by
XBOOLE_1: 4
.= (((L2
\/ (
LSeg (p01,p2)))
\/ (
LSeg (p00,p2)))
\/ (L3
\/ L4)) by
XBOOLE_1: 4
.=
R^2-unit_square by
A41,
TOPREAL1:def 2,
XBOOLE_1: 4;
A43: (P1
/\ P2)
= (((
LSeg (p1,p00))
/\ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p1,p00))
/\ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ (L4
\/ L2))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
A34,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p00))
/\ L4)
\/ ((
LSeg (p1,p00))
/\ L2))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ ((
LSeg (p00,p2))
/\ ((
LSeg (p1,p10))
\/ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ L4)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p00,p2))
/\ ((L4
\/ L2)
\/ (
LSeg (p01,p2)))))) by
A21,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ L4)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
\/ (((
LSeg (p00,p2))
/\ (L4
\/ L2))
\/
{p2}))) by
A40,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ L4)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
\/ ((((
LSeg (p00,p2))
/\ L4)
\/ ((
LSeg (p00,p2))
/\ L2))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p00))
/\ L4)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2}))) by
A35;
A44:
now
per cases ;
suppose
A45: p1
= p00;
A46: p1
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
p1
in (
LSeg (p00,p2)) by
A45,
RLTOPSP1: 68;
then
A47: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
<>
{} by
A46,
XBOOLE_0:def 4;
((
LSeg (p1,p00))
/\ L4)
= (
{p00}
/\ L4) by
A45,
RLTOPSP1: 70;
then
A48: ((
LSeg (p1,p00))
/\ L4)
=
{} by
Lm1,
Lm12;
((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
c=
{p1} by
A18,
A45,
Lm20,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
=
{p1} by
A47,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ (
{p1}
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2})) by
A43,
A48,
XBOOLE_1: 4
.= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2}));
end;
suppose
A49: p1
= p10;
then
A50: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
= ((
LSeg (p00,p2))
/\
{p10}) by
RLTOPSP1: 70;
not p10
in (
LSeg (p00,p2)) by
A26,
Lm4,
Lm6,
Lm8,
TOPREAL1: 3;
then ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A50,
Lm1;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2})) by
A43,
A49,
TOPREAL1: 16,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2}));
end;
suppose
A51: p1
<> p10 & p1
<> p00;
now
assume p00
in ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)));
then
A52: p00
in (
LSeg (p1,p10)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then
0
= (p1
`1 ) by
A14,
A16,
A52,
Lm4,
TOPREAL1: 3;
hence contradiction by
A14,
A17,
A51,
EUCLID: 53;
end;
then
A53:
{p00}
<> ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
c=
{p00} by
A12,
A26,
TOPREAL1: 17,
XBOOLE_1: 27;
then
A54: ((
LSeg (p00,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A53,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p00))
/\ L4);
then
A55: p10
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p10
`1 )
<= (p1
`1 ) by
A55,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A14,
A15,
Lm8,
XXREAL_0: 1;
hence contradiction by
A14,
A17,
A51,
EUCLID: 53;
end;
then
A56:
{p10}
<> ((
LSeg (p1,p00))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ L4)
c=
{p10} by
A3,
Lm21,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p00))
/\ L4)
=
{} by
A56,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p00,p2))
/\ L2)
\/
{p2})) by
A43,
A54;
end;
end;
now
per cases ;
suppose
A57: p2
= p00;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then
A58: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
<>
{} by
A57,
Lm20,
XBOOLE_0:def 4;
((
LSeg (p00,p2))
/\ L2)
= (
{p00}
/\ L2) by
A57,
RLTOPSP1: 70;
then
A59: ((
LSeg (p00,p2))
/\ L2)
=
{} by
Lm1,
Lm13;
((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
c= (L3
/\ L1) by
A10,
A36,
XBOOLE_1: 27;
then ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
=
{p2} by
A57,
A58,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A44,
A59,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
hence thesis;
end;
suppose
A60: p2
= p01;
then
A61: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
= ((
LSeg (p1,p00))
/\
{p01}) by
RLTOPSP1: 70;
not p01
in (
LSeg (p1,p00)) by
A10,
Lm5,
Lm7,
Lm9,
TOPREAL1: 4;
then
A62: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
=
{} by
A61,
Lm1;
hence thesis by
A44,
A60,
ENUMSET1: 1,
TOPREAL1: 15;
thus thesis by
A44,
A60,
A62,
ENUMSET1: 1,
TOPREAL1: 15;
end;
suppose
A63: p2
<> p01 & p2
<> p00;
now
assume p01
in ((
LSeg (p00,p2))
/\ L2);
then
A64: p01
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p2
`2 ) by
A28,
EUCLID: 52;
then (p01
`2 )
<= (p2
`2 ) by
A64,
TOPREAL1: 4;
then 1
= (p2
`2 ) by
A28,
Lm7,
XXREAL_0: 1;
hence contradiction by
A28,
A63,
EUCLID: 53;
end;
then
A65:
{p01}
<> ((
LSeg (p00,p2))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p00,p2))
/\ L2)
c=
{p01} by
A18,
Lm20,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
then
A66: ((
LSeg (p00,p2))
/\ L2)
=
{} by
A65,
ZFMISC_1: 33;
now
assume p00
in ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)));
then
A67: p00
in (
LSeg (p2,p01)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A28,
EUCLID: 52;
then (p2
`2 )
=
0 by
A28,
A67,
Lm5,
TOPREAL1: 4;
hence contradiction by
A28,
A63,
EUCLID: 53;
end;
then
A68:
{p00}
<> ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
c= (L3
/\ L1) by
A10,
A36,
XBOOLE_1: 27;
then
A69: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
=
{} by
A68,
TOPREAL1: 17,
ZFMISC_1: 33;
hence thesis by
A44,
A66,
ENUMSET1: 1;
thus thesis by
A44,
A69,
A66,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A70: p2
in L2;
then
A71: (
LSeg (p2,p11))
c= L2 by
Lm26,
TOPREAL1: 6;
then
A72: ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
=
{} by
A12,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A73: (
LSeg (p2,p01))
c= L2 by
A70,
Lm23,
TOPREAL1: 6;
then
A74: ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))
=
{} by
A10,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
take P1 = (((
LSeg (p1,p00))
\/ L1)
\/ (
LSeg (p01,p2))), P2 = (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2)));
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A75: (L1
/\ (
LSeg (p01,p2)))
<>
{} by
Lm22,
XBOOLE_0:def 4;
(L1
/\ (
LSeg (p01,p2)))
c=
{p01} by
A70,
Lm23,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
then (L1
/\ (
LSeg (p01,p2)))
=
{p01} by
A75,
ZFMISC_1: 33;
then
A76: (L1
\/ (
LSeg (p01,p2)))
is_an_arc_of (p00,p2) by
Lm5,
Lm7,
TOPREAL1: 12;
((
LSeg (p1,p00))
/\ (L1
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p00))
/\ L1)
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.=
{p00} by
A8,
A6,
A74,
TOPREAL1: 17,
ZFMISC_1: 33;
then ((
LSeg (p1,p00))
\/ (L1
\/ (
LSeg (p01,p2))))
is_an_arc_of (p1,p2) by
A76,
TOPREAL1: 11;
hence P1
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A77: (L4
/\ (
LSeg (p11,p2)))
<>
{} by
Lm27,
XBOOLE_0:def 4;
(L4
/\ (
LSeg (p11,p2)))
c= (L4
/\ L2) by
A70,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then (L4
/\ (
LSeg (p11,p2)))
=
{p11} by
A77,
TOPREAL1: 18,
ZFMISC_1: 33;
then
A78: (L4
\/ (
LSeg (p11,p2)))
is_an_arc_of (p10,p2) by
Lm9,
Lm11,
TOPREAL1: 12;
((
LSeg (p1,p10))
/\ (L4
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p10))
/\ L4)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.=
{p10} by
A5,
A7,
A72,
TOPREAL1: 16,
ZFMISC_1: 33;
then ((
LSeg (p1,p10))
\/ (L4
\/ (
LSeg (p11,p2))))
is_an_arc_of (p1,p2) by
A78,
TOPREAL1: 11;
hence P2
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
thus
R^2-unit_square
= ((L1
\/ ((
LSeg (p01,p2))
\/ (
LSeg (p11,p2))))
\/ (L3
\/ L4)) by
A70,
TOPREAL1: 5,
TOPREAL1:def 2
.= (((L1
\/ (
LSeg (p01,p2)))
\/ (
LSeg (p11,p2)))
\/ (L3
\/ L4)) by
XBOOLE_1: 4
.= ((L1
\/ (
LSeg (p01,p2)))
\/ ((L3
\/ L4)
\/ (
LSeg (p11,p2)))) by
XBOOLE_1: 4
.= ((L1
\/ (
LSeg (p01,p2)))
\/ (L3
\/ (L4
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 4
.= ((L1
\/ (
LSeg (p01,p2)))
\/ (((
LSeg (p1,p00))
\/ (
LSeg (p1,p10)))
\/ (L4
\/ (
LSeg (p11,p2))))) by
A3,
TOPREAL1: 5
.= ((L1
\/ (
LSeg (p01,p2)))
\/ ((
LSeg (p1,p00))
\/ ((
LSeg (p1,p10))
\/ (L4
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 4
.= ((L1
\/ (
LSeg (p01,p2)))
\/ ((
LSeg (p1,p00))
\/ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 4
.= (((
LSeg (p1,p00))
\/ (L1
\/ (
LSeg (p01,p2))))
\/ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2)))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A79: ex q st q
= p2 & (q
`1 )
<= 1 & (q
`1 )
>=
0 & (q
`2 )
= 1 by
A70,
TOPREAL1: 13;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A80: ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
=
{} by
A10,
A71,
XBOOLE_1: 3,
XBOOLE_1: 27;
A81: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A12,
A73,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 27;
A82: ((
LSeg (p01,p2))
/\ (
LSeg (p11,p2)))
=
{p2} by
A70,
TOPREAL1: 8;
A83: (P1
/\ P2)
= ((((
LSeg (p1,p00))
\/ L1)
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))
\/ ((
LSeg (p01,p2))
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
\/ L1)
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))
\/ (((
LSeg (p01,p2))
/\ ((
LSeg (p1,p10))
\/ L4))
\/
{p2})) by
A82,
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
\/ L1)
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))
\/ ((((
LSeg (p01,p2))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p01,p2))
/\ L4))
\/
{p2})) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p00))
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2))))
\/ (L1
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
A81,
XBOOLE_1: 23
.= (((((
LSeg (p1,p00))
/\ ((
LSeg (p1,p10))
\/ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2))))
\/ (L1
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L4))
\/ (L1
/\ (((
LSeg (p1,p10))
\/ L4)
\/ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
A9,
A80,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((L1
/\ ((
LSeg (p1,p10))
\/ L4))
\/ (L1
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L4))
\/ (((L1
/\ (
LSeg (p1,p10)))
\/ (L1
/\ L4))
\/ (L1
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p00))
/\ L4))
\/ ((L1
/\ (
LSeg (p1,p10)))
\/ (L1
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
Lm3;
A84:
now
per cases ;
suppose
A85: p1
= p00;
then ((
LSeg (p1,p00))
/\ L4)
= (
{p00}
/\ L4) by
RLTOPSP1: 70;
then ((
LSeg (p1,p00))
/\ L4)
=
{} by
Lm1,
Lm12;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ (L1
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
A83,
A85,
TOPREAL1: 17,
XBOOLE_1: 4
.= ((
{p1}
\/ (L1
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2}));
end;
suppose
A86: p1
= p10;
then (L1
/\ (
LSeg (p1,p10)))
= (L1
/\
{p10}) by
RLTOPSP1: 70;
then (L1
/\ (
LSeg (p1,p10)))
=
{} by
Lm1,
Lm16;
hence (P1
/\ P2)
= ((
{p1}
\/ (L1
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
A83,
A86,
TOPREAL1: 16;
end;
suppose
A87: p1
<> p10 & p1
<> p00;
now
assume p00
in (L1
/\ (
LSeg (p1,p10)));
then
A88: p00
in (
LSeg (p1,p10)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then
0
= (p1
`1 ) by
A14,
A16,
A88,
Lm4,
TOPREAL1: 3;
hence contradiction by
A14,
A17,
A87,
EUCLID: 53;
end;
then
A89:
{p00}
<> (L1
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
(L1
/\ (
LSeg (p1,p10)))
c=
{p00} by
A3,
Lm24,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then
A90: (L1
/\ (
LSeg (p1,p10)))
=
{} by
A89,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p00))
/\ L4);
then
A91: p10
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p10
`1 )
<= (p1
`1 ) by
A91,
TOPREAL1: 3;
then 1
= (p1
`1 ) by
A14,
A15,
Lm8,
XXREAL_0: 1;
hence contradiction by
A14,
A17,
A87,
EUCLID: 53;
end;
then
A92:
{p10}
<> ((
LSeg (p1,p00))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ L4)
c=
{p10} by
A3,
Lm21,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p00))
/\ L4)
=
{} by
A92,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ (L1
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p01,p2))
/\ L4)
\/
{p2})) by
A83,
A90;
end;
end;
now
per cases ;
suppose
A93: p2
= p01;
then ((
LSeg (p01,p2))
/\ L4)
= (
{p01}
/\ L4) by
RLTOPSP1: 70;
then ((
LSeg (p01,p2))
/\ L4)
=
{} by
Lm1,
Lm15;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A84,
A93,
TOPREAL1: 15,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A94: p2
= p11;
then (L1
/\ (
LSeg (p11,p2)))
= (L1
/\
{p11}) by
RLTOPSP1: 70;
then (L1
/\ (
LSeg (p11,p2)))
=
{} by
Lm1,
Lm18;
hence thesis by
A84,
A94,
ENUMSET1: 1,
TOPREAL1: 18;
end;
suppose
A95: p2
<> p11 & p2
<> p01;
now
assume p11
in ((
LSeg (p01,p2))
/\ L4);
then
A96: p11
in (
LSeg (p01,p2)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A79,
EUCLID: 52;
then (p11
`1 )
<= (p2
`1 ) by
A96,
TOPREAL1: 3;
then 1
= (p2
`1 ) by
A79,
Lm10,
XXREAL_0: 1;
hence contradiction by
A79,
A95,
EUCLID: 53;
end;
then
A97:
{p11}
<> ((
LSeg (p01,p2))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ L4)
c=
{p11} by
A70,
Lm23,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
then
A98: ((
LSeg (p01,p2))
/\ L4)
=
{} by
A97,
ZFMISC_1: 33;
now
assume p01
in (L1
/\ (
LSeg (p11,p2)));
then
A99: p01
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A79,
EUCLID: 52;
then (p2
`1 )
=
0 by
A79,
A99,
Lm6,
TOPREAL1: 3;
hence contradiction by
A79,
A95,
EUCLID: 53;
end;
then
A100:
{p01}
<> (L1
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
(L1
/\ (
LSeg (p11,p2)))
c=
{p01} by
A70,
Lm26,
TOPREAL1: 6,
TOPREAL1: 15,
XBOOLE_1: 26;
then (L1
/\ (
LSeg (p11,p2)))
=
{} by
A100,
ZFMISC_1: 33;
hence thesis by
A84,
A98,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A101: p2
in L3;
A102: p
=
|[(p
`1 ), (p
`2 )]| by
EUCLID: 53;
A103: (
LSeg (p1,p2))
c= L3 by
A3,
A101,
TOPREAL1: 6;
consider q such that
A104: q
= p2 and
A105: (q
`1 )
<= 1 and
A106: (q
`1 )
>=
0 and
A107: (q
`2 )
=
0 by
A101,
TOPREAL1: 13;
A108: q
=
|[(q
`1 ), (q
`2 )]| by
EUCLID: 53;
now
per cases by
A1,
A14,
A17,
A104,
A107,
A102,
A108,
XXREAL_0: 1;
suppose
A109: (p
`1 )
< (q
`1 );
now
assume p10
in ((
LSeg (p1,p00))
/\ L4);
then
A110: p10
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p10
`1 )
<= (p1
`1 ) by
A110,
TOPREAL1: 3;
hence contradiction by
A14,
A15,
A105,
A109,
Lm8,
XXREAL_0: 1;
end;
then
A111:
{p10}
<> ((
LSeg (p1,p00))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p00))
/\ L4)
c=
{p10} by
A3,
Lm21,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then
A112: ((
LSeg (p1,p00))
/\ L4)
=
{} by
A111,
ZFMISC_1: 33;
p00
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
then
A113: ((
LSeg (p1,p00))
/\ L1)
<>
{} by
Lm20,
XBOOLE_0:def 4;
now
assume p00
in (L1
/\ (
LSeg (p10,p2)));
then
A114: p00
in (
LSeg (p2,p10)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A104,
A105,
EUCLID: 52;
hence contradiction by
A16,
A104,
A109,
A114,
Lm4,
TOPREAL1: 3;
end;
then
A115:
{p00}
<> (L1
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
(L1
/\ (
LSeg (p10,p2)))
c=
{p00} by
A101,
Lm24,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then
A116: (L1
/\ (
LSeg (p10,p2)))
=
{} by
A115,
ZFMISC_1: 33;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A117: ((
LSeg (p1,p2))
/\ L2)
=
{} by
A103,
XBOOLE_1: 3,
XBOOLE_1: 26;
A118: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
c=
{p1}
proof
let a be
object;
assume
A119: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A120: p
in (
LSeg (p00,p1)) by
A119,
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then
A121: (p
`1 )
<= (p1
`1 ) by
A120,
TOPREAL1: 3;
A122: p
in (
LSeg (p1,p2)) by
A119,
XBOOLE_0:def 4;
then (p1
`1 )
<= (p
`1 ) by
A14,
A104,
A109,
TOPREAL1: 3;
then
A123: (p1
`1 )
= (p
`1 ) by
A121,
XXREAL_0: 1;
(p1
`2 )
<= (p
`2 ) by
A14,
A17,
A104,
A107,
A122,
TOPREAL1: 4;
then (p
`2 )
=
0 by
A14,
A17,
A104,
A107,
A122,
TOPREAL1: 4;
then p
=
|[(p1
`1 ),
0 ]| by
A123,
EUCLID: 53
.= p1 by
A14,
A17,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A124: ((
LSeg (p1,p00))
/\ L1)
c= (L3
/\ L1) by
A3,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p00))
\/ (((L1
\/ L2)
\/ L4)
\/ (
LSeg (p10,p2))));
A125: (L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
(L2
/\ (
LSeg (p10,p2)))
c= (L2
/\ L3) by
A101,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A126: (L2
/\ (
LSeg (p10,p2)))
=
{} by
A125,
XBOOLE_1: 3;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A127: ((L1
\/ L2)
/\ L4)
= ((L1
/\ L4)
\/ (L2
/\ L4)) by
XBOOLE_1: 23
.=
{p11} by
Lm3,
TOPREAL1: 18;
A128: ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)))
c=
{p2}
proof
let a be
object;
assume
A129: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A130: p
in (
LSeg (p2,p10)) by
A129,
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A104,
A105,
EUCLID: 52;
then
A131: (p2
`1 )
<= (p
`1 ) by
A130,
TOPREAL1: 3;
A132: p
in (
LSeg (p1,p2)) by
A129,
XBOOLE_0:def 4;
then (p
`1 )
<= (p2
`1 ) by
A14,
A104,
A109,
TOPREAL1: 3;
then
A133: (p2
`1 )
= (p
`1 ) by
A131,
XXREAL_0: 1;
(p1
`2 )
<= (p
`2 ) by
A14,
A17,
A104,
A107,
A132,
TOPREAL1: 4;
then (p
`2 )
=
0 by
A14,
A17,
A104,
A107,
A132,
TOPREAL1: 4;
then p
=
|[(p2
`1 ),
0 ]| by
A133,
EUCLID: 53
.= p2 by
A104,
A107,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A134: ((
LSeg (p1,p00))
/\ L2)
=
{} by
A10,
XBOOLE_1: 3,
XBOOLE_1: 26;
A135:
now
set a = the
Element of ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)));
assume
A136: ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A137: p
in (
LSeg (p00,p1)) by
A136,
XBOOLE_0:def 4;
A138: p
in (
LSeg (p2,p10)) by
A136,
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A104,
A105,
EUCLID: 52;
then
A139: (p2
`1 )
<= (p
`1 ) by
A138,
TOPREAL1: 3;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p
`1 )
<= (p1
`1 ) by
A137,
TOPREAL1: 3;
hence contradiction by
A14,
A104,
A109,
A139,
XXREAL_0: 2;
end;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A140: (L4
/\ (
LSeg (p10,p2)))
<>
{} by
Lm25,
XBOOLE_0:def 4;
(L4
/\ (
LSeg (p10,p2)))
c= (L4
/\ L3) by
A101,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A141: (L4
/\ (
LSeg (p10,p2)))
=
{p10} by
A140,
TOPREAL1: 16,
ZFMISC_1: 33;
(L1
\/ L2)
is_an_arc_of (p00,p11) by
Lm5,
Lm7,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 15;
then
A142: ((L1
\/ L2)
\/ L4)
is_an_arc_of (p00,p10) by
A127,
TOPREAL1: 10;
(((L1
\/ L2)
\/ L4)
/\ (
LSeg (p10,p2)))
= (((L1
\/ L2)
/\ (
LSeg (p10,p2)))
\/ (L4
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((L1
/\ (
LSeg (p10,p2)))
\/ (L2
/\ (
LSeg (p10,p2))))
\/
{p10}) by
A141,
XBOOLE_1: 23
.=
{p10} by
A116,
A126;
then
A143: (((L1
\/ L2)
\/ L4)
\/ (
LSeg (p10,p2)))
is_an_arc_of (p00,p2) by
A142,
TOPREAL1: 10;
((
LSeg (p1,p00))
/\ (((L1
\/ L2)
\/ L4)
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p00))
/\ ((L1
\/ L2)
\/ L4))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p00))
/\ (L1
\/ L2))
\/ ((
LSeg (p1,p00))
/\ L4)) by
A135,
XBOOLE_1: 23
.= (((
LSeg (p1,p00))
/\ L1)
\/ ((
LSeg (p1,p00))
/\ L2)) by
A112,
XBOOLE_1: 23
.=
{p00} by
A134,
A124,
A113,
TOPREAL1: 17,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A143,
TOPREAL1: 11;
A144: p1
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00))) by
A144,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00))) by
ZFMISC_1: 31;
then
A145: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
=
{p1} by
A118,
XBOOLE_0:def 10;
thus (P1
\/ P2)
= (((
LSeg (p00,p1))
\/ (
LSeg (p1,p2)))
\/ (((L1
\/ L2)
\/ L4)
\/ (
LSeg (p10,p2)))) by
XBOOLE_1: 4
.= ((((
LSeg (p00,p1))
\/ (
LSeg (p1,p2)))
\/ (
LSeg (p2,p10)))
\/ ((L1
\/ L2)
\/ L4)) by
XBOOLE_1: 4
.= (((L1
\/ L2)
\/ L4)
\/ L3) by
A3,
A101,
TOPREAL1: 7
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A146: p2
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2))) by
A146,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
then
A147: ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)))
=
{p2} by
A128,
XBOOLE_0:def 10;
A148: (P1
/\ P2)
= (((
LSeg (p1,p2))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p1,p2))
/\ (((L1
\/ L2)
\/ L4)
\/ (
LSeg (p10,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L1
\/ L2)
\/ L4))
\/
{p2})) by
A145,
A147,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L1
\/ L2))
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L1)
\/ ((
LSeg (p1,p2))
/\ L2))
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L1)
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2}))) by
A117,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L1))
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
XBOOLE_1: 4;
A149:
now
per cases ;
suppose
A150: p1
= p00;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then
A151: ((
LSeg (p1,p2))
/\ L1)
<>
{} by
A150,
Lm20,
XBOOLE_0:def 4;
((
LSeg (p1,p2))
/\ L1)
c= (L3
/\ L1) by
A3,
A101,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L1)
=
{p1} by
A150,
A151,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
A148;
end;
suppose
A152: p1
<> p00;
now
assume p00
in ((
LSeg (p1,p2))
/\ L1);
then p00
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p1
`1 )
=
0 by
A14,
A16,
A104,
A109,
Lm4,
TOPREAL1: 3;
hence contradiction by
A14,
A17,
A152,
EUCLID: 53;
end;
then
A153:
{p00}
<> ((
LSeg (p1,p2))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p2))
/\ L1)
c= (L3
/\ L1) by
A3,
A101,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L1)
=
{} by
A153,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/
{p2})) by
A148;
end;
end;
now
per cases ;
suppose
A154: p2
= p10;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then
A155: ((
LSeg (p1,p2))
/\ L4)
<>
{} by
A154,
Lm25,
XBOOLE_0:def 4;
((
LSeg (p1,p2))
/\ L4)
c=
{p2} by
A3,
A101,
A154,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L4)
=
{p2} by
A155,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A149,
ENUMSET1: 1;
end;
suppose
A156: p2
<> p10;
now
assume p10
in ((
LSeg (p1,p2))
/\ L4);
then p10
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p10
`1 )
<= (p2
`1 ) by
A14,
A104,
A109,
TOPREAL1: 3;
then (p2
`1 )
= 1 by
A104,
A105,
Lm8,
XXREAL_0: 1;
hence contradiction by
A104,
A107,
A156,
EUCLID: 53;
end;
then
A157:
{p10}
<> ((
LSeg (p1,p2))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p2))
/\ L4)
c=
{p10} by
A3,
A101,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L4)
=
{} by
A157,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A149,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
suppose
A158: (q
`1 )
< (p
`1 );
A159: ((
LSeg (p1,p2))
/\ (
LSeg (p00,p2)))
c=
{p2}
proof
let a be
object;
assume
A160: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p00,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A161: p
in (
LSeg (p00,p2)) by
A160,
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A104,
A106,
EUCLID: 52;
then
A162: (p
`1 )
<= (p2
`1 ) by
A161,
TOPREAL1: 3;
A163: p
in (
LSeg (p2,p1)) by
A160,
XBOOLE_0:def 4;
then (p2
`1 )
<= (p
`1 ) by
A14,
A104,
A158,
TOPREAL1: 3;
then
A164: (p2
`1 )
= (p
`1 ) by
A162,
XXREAL_0: 1;
(p2
`2 )
<= (p
`2 ) by
A14,
A17,
A104,
A107,
A163,
TOPREAL1: 4;
then (p
`2 )
=
0 by
A14,
A17,
A104,
A107,
A163,
TOPREAL1: 4;
then p
=
|[(p2
`1 ),
0 ]| by
A164,
EUCLID: 53
.= p2 by
A104,
A107,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A165: ((
LSeg (p1,p10))
/\ L4)
<>
{} by
Lm25,
XBOOLE_0:def 4;
now
assume p10
in (L4
/\ (
LSeg (p00,p2)));
then
A166: p10
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A104,
A106,
EUCLID: 52;
then (p10
`1 )
<= (p2
`1 ) by
A166,
TOPREAL1: 3;
hence contradiction by
A15,
A104,
A105,
A158,
Lm8,
XXREAL_0: 1;
end;
then
A167:
{p10}
<> (L4
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
(L4
/\ (
LSeg (p00,p2)))
c= (L4
/\ L3) by
A101,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A168: (L4
/\ (
LSeg (p00,p2)))
=
{} by
A167,
TOPREAL1: 16,
ZFMISC_1: 33;
A169: ((L4
\/ L2)
/\ L1)
= ((L1
/\ L4)
\/ (L2
/\ L1)) by
XBOOLE_1: 23
.=
{p01} by
Lm3,
TOPREAL1: 15;
(L4
\/ L2)
is_an_arc_of (p10,p01) by
Lm9,
Lm11,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 18;
then
A170: ((L4
\/ L2)
\/ L1)
is_an_arc_of (p10,p00) by
A169,
TOPREAL1: 10;
now
assume p00
in ((
LSeg (p1,p10))
/\ L1);
then
A171: p00
in (
LSeg (p1,p10)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
hence contradiction by
A14,
A106,
A158,
A171,
Lm4,
TOPREAL1: 3;
end;
then
A172:
{p00}
<> ((
LSeg (p1,p10))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ L1)
c= (L3
/\ L1) by
A3,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A173: ((
LSeg (p1,p10))
/\ L1)
=
{} by
A172,
TOPREAL1: 17,
ZFMISC_1: 33;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A174: (L1
/\ (
LSeg (p00,p2)))
<>
{} by
Lm20,
XBOOLE_0:def 4;
(L1
/\ (
LSeg (p00,p2)))
c=
{p00} by
A101,
Lm21,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then
A175: (L1
/\ (
LSeg (p00,p2)))
=
{p00} by
A174,
ZFMISC_1: 33;
A176: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)))
c=
{p1}
proof
let a be
object;
assume
A177: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A178: p
in (
LSeg (p1,p10)) by
A177,
XBOOLE_0:def 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then
A179: (p1
`1 )
<= (p
`1 ) by
A178,
TOPREAL1: 3;
A180: p
in (
LSeg (p2,p1)) by
A177,
XBOOLE_0:def 4;
then (p
`1 )
<= (p1
`1 ) by
A14,
A104,
A158,
TOPREAL1: 3;
then
A181: (p1
`1 )
= (p
`1 ) by
A179,
XXREAL_0: 1;
(p2
`2 )
<= (p
`2 ) by
A14,
A17,
A104,
A107,
A180,
TOPREAL1: 4;
then (p
`2 )
=
0 by
A14,
A17,
A104,
A107,
A180,
TOPREAL1: 4;
then p
=
|[(p1
`1 ),
0 ]| by
A181,
EUCLID: 53
.= p1 by
A14,
A17,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A182: ((
LSeg (p1,p10))
/\ L2)
=
{} by
A12,
XBOOLE_1: 3,
XBOOLE_1: 26;
A183:
now
set a = the
Element of ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)));
assume
A184: ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A185: p
in (
LSeg (p1,p10)) by
A184,
XBOOLE_0:def 4;
A186: p
in (
LSeg (p00,p2)) by
A184,
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A104,
A106,
EUCLID: 52;
then
A187: (p
`1 )
<= (p2
`1 ) by
A186,
TOPREAL1: 3;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then (p1
`1 )
<= (p
`1 ) by
A185,
TOPREAL1: 3;
hence contradiction by
A14,
A104,
A158,
A187,
XXREAL_0: 2;
end;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p10))
\/ (((L4
\/ L2)
\/ L1)
\/ (
LSeg (p00,p2))));
A188: (L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
(L2
/\ (
LSeg (p00,p2)))
c= (L2
/\ L3) by
A101,
Lm21,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A189: (L2
/\ (
LSeg (p00,p2)))
=
{} by
A188,
XBOOLE_1: 3;
A190: ((
LSeg (p1,p10))
/\ L4)
c=
{p10} by
A3,
Lm24,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
(((L4
\/ L2)
\/ L1)
/\ (
LSeg (p00,p2)))
= (((L4
\/ L2)
/\ (
LSeg (p00,p2)))
\/ (L1
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.= (((L4
/\ (
LSeg (p00,p2)))
\/ (L2
/\ (
LSeg (p00,p2))))
\/
{p00}) by
A175,
XBOOLE_1: 23
.=
{p00} by
A168,
A189;
then
A191: (((L4
\/ L2)
\/ L1)
\/ (
LSeg (p00,p2)))
is_an_arc_of (p10,p2) by
A170,
TOPREAL1: 10;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A192: p2
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
((
LSeg (p1,p10))
/\ (((L4
\/ L2)
\/ L1)
\/ (
LSeg (p00,p2))))
= (((
LSeg (p1,p10))
/\ ((L4
\/ L2)
\/ L1))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ (L4
\/ L2))
\/ ((
LSeg (p1,p10))
/\ L1)) by
A183,
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ L4)
\/ ((
LSeg (p1,p10))
/\ L2)) by
A173,
XBOOLE_1: 23
.=
{p10} by
A182,
A190,
A165,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A191,
TOPREAL1: 11;
A193: p1
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
thus (P1
\/ P2)
= (((
LSeg (p2,p1))
\/ (
LSeg (p1,p10)))
\/ (((L4
\/ L2)
\/ L1)
\/ (
LSeg (p00,p2)))) by
XBOOLE_1: 4
.= (((
LSeg (p00,p2))
\/ ((
LSeg (p2,p1))
\/ (
LSeg (p1,p10))))
\/ ((L4
\/ L2)
\/ L1)) by
XBOOLE_1: 4
.= (L3
\/ ((L4
\/ L2)
\/ L1)) by
A3,
A101,
TOPREAL1: 7
.= (L3
\/ (L4
\/ (L1
\/ L2))) by
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A194: ((
LSeg (p1,p2))
/\ L2)
=
{} by
A103,
XBOOLE_1: 3,
XBOOLE_1: 26;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p00,p2))) by
A192,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
then
A195: ((
LSeg (p1,p2))
/\ (
LSeg (p00,p2)))
=
{p2} by
A159,
XBOOLE_0:def 10;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10))) by
A193,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)))
=
{p1} by
A176,
XBOOLE_0:def 10;
then
A196: (P1
/\ P2)
= (
{p1}
\/ ((
LSeg (p1,p2))
/\ (((L4
\/ L2)
\/ L1)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L4
\/ L2)
\/ L1))
\/
{p2})) by
A195,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L4
\/ L2))
\/ ((
LSeg (p1,p2))
/\ L1))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L4)
\/ ((
LSeg (p1,p2))
/\ L2))
\/ ((
LSeg (p1,p2))
/\ L1))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L4)
\/ (((
LSeg (p1,p2))
/\ L1)
\/
{p2}))) by
A194,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/ (((
LSeg (p1,p2))
/\ L1)
\/
{p2})) by
XBOOLE_1: 4;
A197:
now
per cases ;
suppose
A198: p2
= p00;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then
A199: ((
LSeg (p1,p2))
/\ L1)
<>
{} by
A198,
Lm20,
XBOOLE_0:def 4;
((
LSeg (p1,p2))
/\ L1)
c= (L3
/\ L1) by
A3,
A101,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L1)
=
{p2} by
A198,
A199,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2}) by
A196;
end;
suppose
A200: p2
<> p00;
now
assume p00
in ((
LSeg (p1,p2))
/\ L1);
then p00
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p2
`1 )
=
0 by
A14,
A104,
A106,
A158,
Lm4,
TOPREAL1: 3;
hence contradiction by
A104,
A107,
A200,
EUCLID: 53;
end;
then
A201:
{p00}
<> ((
LSeg (p1,p2))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p2))
/\ L1)
c= (L3
/\ L1) by
A3,
A101,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L1)
=
{} by
A201,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L4))
\/
{p2}) by
A196;
end;
end;
now
per cases ;
suppose
A202: p1
= p10;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then
A203: ((
LSeg (p1,p2))
/\ L4)
<>
{} by
A202,
Lm25,
XBOOLE_0:def 4;
((
LSeg (p1,p2))
/\ L4)
c=
{p1} by
A3,
A101,
A202,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L4)
=
{p1} by
A203,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A197,
ENUMSET1: 1;
end;
suppose
A204: p1
<> p10;
now
assume p10
in ((
LSeg (p1,p2))
/\ L4);
then p10
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p10
`1 )
<= (p1
`1 ) by
A14,
A104,
A158,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A14,
A15,
Lm8,
XXREAL_0: 1;
hence contradiction by
A14,
A17,
A204,
EUCLID: 53;
end;
then
A205:
{p10}
<> ((
LSeg (p1,p2))
/\ L4) by
ZFMISC_1: 31;
((
LSeg (p1,p2))
/\ L4)
c=
{p10} by
A3,
A101,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then ((
LSeg (p1,p2))
/\ L4)
=
{} by
A205,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A197,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
end;
hence thesis;
end;
suppose
A206: p2
in L4;
then
A207: ex q st q
= p2 & (q
`1 )
= 1 & (q
`2 )
<= 1 & (q
`2 )
>=
0 by
TOPREAL1: 13;
now
assume
A208: p10
in ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)));
then
A209: p10
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p10
`1 )
<= (p1
`1 ) by
A209,
TOPREAL1: 3;
then 1
= (p1
`1 ) by
A14,
A15,
Lm8,
XXREAL_0: 1;
then
A210: p1
= p10 by
A14,
A17,
EUCLID: 53;
A211: (p2
`2 )
<= (p11
`2 ) by
A207,
EUCLID: 52;
p10
in (
LSeg (p2,p11)) by
A208,
XBOOLE_0:def 4;
then
0
= (p2
`2 ) by
A207,
A211,
Lm9,
TOPREAL1: 4;
hence contradiction by
A1,
A207,
A210,
EUCLID: 53;
end;
then
A212:
{p10}
<> ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
A213: L2
is_an_arc_of (p01,p11) by
Lm6,
Lm10,
TOPREAL1: 9;
L1
is_an_arc_of (p00,p01) by
Lm5,
Lm7,
TOPREAL1: 9;
then
A214: (L1
\/ L2)
is_an_arc_of (p00,p11) by
A213,
TOPREAL1: 2,
TOPREAL1: 15;
take P1 = ((
LSeg (p1,p10))
\/ (
LSeg (p10,p2))), P2 = ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2))));
A215: L3
= ((
LSeg (p1,p10))
\/ (
LSeg (p1,p00))) by
A3,
TOPREAL1: 5;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A216: (L2
/\ (
LSeg (p11,p2)))
<>
{} by
Lm26,
XBOOLE_0:def 4;
A217: p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A218: ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
<>
{} by
A217,
XBOOLE_0:def 4;
A219: (
LSeg (p2,p10))
c= L4 by
A206,
Lm25,
TOPREAL1: 6;
then ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
c= (L3
/\ L4) by
A12,
XBOOLE_1: 27;
then
A220: ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
=
{p10} by
A218,
TOPREAL1: 16,
ZFMISC_1: 33;
p1
<> p10 or p2
<> p10 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A220,
TOPREAL1: 12;
A221: ((
LSeg (p1,p10))
/\ (
LSeg (p1,p00)))
=
{p1} by
A3,
TOPREAL1: 8;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A222: ((
LSeg (p10,p2))
/\ L1)
=
{} by
A219,
XBOOLE_1: 3,
XBOOLE_1: 26;
A223: (
LSeg (p2,p11))
c= L4 by
A206,
Lm27,
TOPREAL1: 6;
then
A224: (L2
/\ (
LSeg (p11,p2)))
c=
{p11} by
TOPREAL1: 18,
XBOOLE_1: 27;
A225: (L1
/\ (
LSeg (p11,p2)))
=
{} by
A223,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L1
\/ L2)
/\ (
LSeg (p11,p2)))
= ((L1
/\ (
LSeg (p11,p2)))
\/ (L2
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.=
{p11} by
A225,
A224,
A216,
ZFMISC_1: 33;
then
A226: ((L1
\/ L2)
\/ (
LSeg (p11,p2)))
is_an_arc_of (p00,p2) by
A214,
TOPREAL1: 10;
A227: ((
LSeg (p10,p2))
/\ (
LSeg (p11,p2)))
=
{p2} by
A206,
TOPREAL1: 8;
A228: L4
= ((
LSeg (p11,p2))
\/ (
LSeg (p10,p2))) by
A206,
TOPREAL1: 5;
((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
c=
{p10} by
A10,
A223,
TOPREAL1: 16,
XBOOLE_1: 27;
then
A229: ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))
=
{} by
A212,
ZFMISC_1: 33;
((
LSeg (p1,p00))
/\ ((L1
\/ L2)
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p00))
/\ (L1
\/ L2))
\/ ((
LSeg (p1,p00))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p00))
/\ L1)
\/ ((
LSeg (p1,p00))
/\ L2)) by
A229,
XBOOLE_1: 23
.=
{p00} by
A8,
A6,
A11,
TOPREAL1: 17,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A226,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p10,p2))
\/ ((
LSeg (p1,p10))
\/ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 4
.= ((
LSeg (p10,p2))
\/ (L3
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2))))) by
A215,
XBOOLE_1: 4
.= ((((L1
\/ L2)
\/ L3)
\/ (
LSeg (p11,p2)))
\/ (
LSeg (p10,p2))) by
XBOOLE_1: 4
.= (((L1
\/ L2)
\/ L3)
\/ L4) by
A228,
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A230: (P1
/\ P2)
= (((
LSeg (p1,p10))
/\ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p10))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p1,p10))
/\ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ (L1
\/ L2))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
A221,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p10))
/\ L1)
\/ ((
LSeg (p10,p1))
/\ L2))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p00))
\/ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L1)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
\/ ((
LSeg (p10,p2))
/\ ((L1
\/ L2)
\/ (
LSeg (p11,p2)))))) by
A13,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L1)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
\/ (((
LSeg (p10,p2))
/\ (L1
\/ L2))
\/
{p2}))) by
A227,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L1)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
\/ ((((
LSeg (p10,p2))
/\ L1)
\/ ((
LSeg (p10,p2))
/\ L2))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L1)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2}))) by
A222;
A231:
now
per cases ;
suppose
A232: p1
= p00;
then
A233: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
= ((
LSeg (p10,p2))
/\
{p00}) by
RLTOPSP1: 70;
not p00
in (
LSeg (p10,p2)) by
A219,
Lm4,
Lm8,
Lm10,
TOPREAL1: 3;
then ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
=
{} by
A233,
Lm1;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2})) by
A230,
A232,
TOPREAL1: 17,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2}));
end;
suppose
A234: p1
= p10;
A235: p1
in (
LSeg (p1,p00)) by
RLTOPSP1: 68;
p1
in (
LSeg (p10,p2)) by
A234,
RLTOPSP1: 68;
then
A236:
{}
<> ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00))) by
A235,
XBOOLE_0:def 4;
((
LSeg (p1,p10))
/\ L1)
= (
{p10}
/\ L1) by
A234,
RLTOPSP1: 70;
then
A237: ((
LSeg (p1,p10))
/\ L1)
=
{} by
Lm1,
Lm16;
((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
c= (L4
/\ L3) by
A10,
A219,
XBOOLE_1: 27;
then ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
=
{p1} by
A234,
A236,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ (
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2})) by
A230,
A237,
XBOOLE_1: 4
.= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2}));
end;
suppose
A238: p1
<> p10 & p1
<> p00;
now
assume p10
in ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)));
then
A239: p10
in (
LSeg (p00,p1)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p1
`1 ) by
A14,
A16,
EUCLID: 52;
then (p10
`1 )
<= (p1
`1 ) by
A239,
TOPREAL1: 3;
then (p1
`1 )
= 1 by
A14,
A15,
Lm8,
XXREAL_0: 1;
hence contradiction by
A14,
A17,
A238,
EUCLID: 53;
end;
then
A240:
{p10}
<> ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00))) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
c= (L4
/\ L3) by
A10,
A219,
XBOOLE_1: 27;
then
A241: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p00)))
=
{} by
A240,
TOPREAL1: 16,
ZFMISC_1: 33;
now
assume p00
in ((
LSeg (p1,p10))
/\ L1);
then
A242: p00
in (
LSeg (p1,p10)) by
XBOOLE_0:def 4;
(p1
`1 )
<= (p10
`1 ) by
A14,
A15,
EUCLID: 52;
then (p1
`1 )
=
0 by
A14,
A16,
A242,
Lm4,
TOPREAL1: 3;
hence contradiction by
A14,
A17,
A238,
EUCLID: 53;
end;
then
A243:
{p00}
<> ((
LSeg (p1,p10))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ L1)
c= (L3
/\ L1) by
A3,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p10))
/\ L1)
=
{} by
A243,
TOPREAL1: 17,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))))
\/ (((
LSeg (p10,p2))
/\ L2)
\/
{p2})) by
A230,
A241;
end;
end;
now
per cases ;
suppose
A244: p2
= p10;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A245:
{}
<> ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))) by
A244,
Lm25,
XBOOLE_0:def 4;
((
LSeg (p10,p2))
/\ L2)
= (
{p10}
/\ L2) by
A244,
RLTOPSP1: 70;
then
A246: ((
LSeg (p10,p2))
/\ L2)
=
{} by
Lm1,
Lm17;
((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
c=
{p2} by
A12,
A244,
TOPREAL1: 16,
XBOOLE_1: 27;
then ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
=
{p2} by
A245,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A231,
A246,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A247: p2
= p11;
then
A248: ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
= ((
LSeg (p1,p10))
/\
{p11}) by
RLTOPSP1: 70;
not p11
in (
LSeg (p1,p10)) by
A12,
Lm5,
Lm9,
Lm11,
TOPREAL1: 4;
then ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
=
{} by
A248,
Lm1;
hence thesis by
A231,
A247,
ENUMSET1: 1,
TOPREAL1: 18;
end;
suppose
A249: p2
<> p11 & p2
<> p10;
now
assume p11
in ((
LSeg (p10,p2))
/\ L2);
then
A250: p11
in (
LSeg (p10,p2)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A207,
EUCLID: 52;
then (p11
`2 )
<= (p2
`2 ) by
A250,
TOPREAL1: 4;
then 1
= (p2
`2 ) by
A207,
Lm11,
XXREAL_0: 1;
hence contradiction by
A207,
A249,
EUCLID: 53;
end;
then
A251:
{p11}
<> ((
LSeg (p10,p2))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ L2)
c= (L4
/\ L2) by
A206,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A252: ((
LSeg (p10,p2))
/\ L2)
=
{} by
A251,
TOPREAL1: 18,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)));
then
A253: p10
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A207,
EUCLID: 52;
then (p2
`2 )
=
0 by
A207,
A253,
Lm9,
TOPREAL1: 4;
hence contradiction by
A207,
A249,
EUCLID: 53;
end;
then
A254:
{p10}
<> ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
c=
{p10} by
A12,
A223,
TOPREAL1: 16,
XBOOLE_1: 27;
then ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
=
{} by
A254,
ZFMISC_1: 33;
hence thesis by
A231,
A252,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
end;
Lm33: p1
<> p2 & p2
in
R^2-unit_square & p1
in (
LSeg (p10,p11)) implies ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) &
R^2-unit_square
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume that
A1: p1
<> p2 and
A2: p2
in
R^2-unit_square and
A3: p1
in (
LSeg (p10,p11));
A4: p2
in (L1
\/ L2) or p2
in (L3
\/ L4) by
A2,
TOPREAL1:def 2,
XBOOLE_0:def 3;
A5: (
LSeg (p1,p11))
c= L4 by
A3,
Lm27,
TOPREAL1: 6;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A6:
{}
<> ((
LSeg (p1,p11))
/\ L2) by
Lm26,
XBOOLE_0:def 4;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A7:
{}
<> ((
LSeg (p1,p10))
/\ L3) by
Lm24,
XBOOLE_0:def 4;
A8: ((
LSeg (p1,p11))
/\ L2)
c= (L4
/\ L2) by
A3,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
A9: ((
LSeg (p1,p10))
/\ L3)
c= (L4
/\ L3) by
A3,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A10: ((
LSeg (p1,p11))
/\ L1)
=
{} by
A5,
XBOOLE_1: 3,
XBOOLE_1: 26;
A11: (
LSeg (p1,p10))
c= L4 by
A3,
Lm25,
TOPREAL1: 6;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A12: ((
LSeg (p10,p1))
/\ L1)
=
{} by
A11,
XBOOLE_1: 3,
XBOOLE_1: 26;
consider p such that
A13: p
= p1 and
A14: (p
`1 )
= 1 and
A15: (p
`2 )
<= 1 and
A16: (p
`2 )
>=
0 by
A3,
TOPREAL1: 13;
per cases by
A4,
XBOOLE_0:def 3;
suppose
A17: p2
in L1;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A18: (L3
/\ (
LSeg (p00,p2)))
<>
{} by
Lm21,
XBOOLE_0:def 4;
(L3
/\ (
LSeg (p00,p2)))
c= (L3
/\ L1) by
A17,
Lm20,
TOPREAL1: 6,
XBOOLE_1: 26;
then (L3
/\ (
LSeg (p00,p2)))
=
{p00} by
A18,
TOPREAL1: 17,
ZFMISC_1: 33;
then
A19: (L3
\/ (
LSeg (p00,p2)))
is_an_arc_of (p10,p2) by
Lm4,
Lm8,
TOPREAL1: 9,
TOPREAL1: 10;
A20: (
LSeg (p2,p00))
c= L1 by
A17,
Lm20,
TOPREAL1: 6;
then
A21: ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
=
{} by
A11,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
A22: (
LSeg (p2,p01))
c= L1 by
A17,
Lm22,
TOPREAL1: 6;
then
A23: ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
=
{} by
A5,
Lm3,
XBOOLE_1: 3,
XBOOLE_1: 27;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A24: ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
=
{} by
A5,
A20,
XBOOLE_1: 3,
XBOOLE_1: 27;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A25: ((
LSeg (p01,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A11,
A22,
XBOOLE_1: 3,
XBOOLE_1: 27;
A26: ((
LSeg (p01,p2))
/\ (
LSeg (p00,p2)))
=
{p2} by
A17,
TOPREAL1: 8;
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A27: p01
in (L2
/\ (
LSeg (p01,p2))) by
Lm23,
XBOOLE_0:def 4;
(L2
/\ (
LSeg (p01,p2)))
c= (L2
/\ L1) by
A17,
Lm22,
TOPREAL1: 6,
XBOOLE_1: 26;
then (L2
/\ (
LSeg (p01,p2)))
=
{p01} by
A27,
TOPREAL1: 15,
ZFMISC_1: 33;
then
A28: (L2
\/ (
LSeg (p01,p2)))
is_an_arc_of (p11,p2) by
Lm6,
Lm10,
TOPREAL1: 9,
TOPREAL1: 10;
take P1 = (((
LSeg (p1,p11))
\/ L2)
\/ (
LSeg (p01,p2))), P2 = (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2)));
A29: ((
LSeg (p1,p11))
\/ (
LSeg (p1,p10)))
= L4 by
A3,
TOPREAL1: 5;
((
LSeg (p1,p11))
/\ (L2
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p11))
/\ L2)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.=
{p11} by
A8,
A6,
A23,
TOPREAL1: 18,
ZFMISC_1: 33;
then ((
LSeg (p1,p11))
\/ (L2
\/ (
LSeg (p01,p2))))
is_an_arc_of (p1,p2) by
A28,
TOPREAL1: 11;
hence P1
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
A30: ex q st q
= p2 & (q
`1 )
=
0 & (q
`2 )
<= 1 & (q
`2 )
>=
0 by
A17,
TOPREAL1: 13;
((
LSeg (p1,p10))
/\ (L3
\/ (
LSeg (p00,p2))))
= (((
LSeg (p1,p10))
/\ L3)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.=
{p10} by
A9,
A7,
A21,
TOPREAL1: 16,
ZFMISC_1: 33;
then ((
LSeg (p1,p10))
\/ (L3
\/ (
LSeg (p00,p2))))
is_an_arc_of (p1,p2) by
A19,
TOPREAL1: 11;
hence P2
is_an_arc_of (p1,p2) by
XBOOLE_1: 4;
thus
R^2-unit_square
= ((((
LSeg (p00,p2))
\/ (
LSeg (p01,p2)))
\/ L2)
\/ (L3
\/ L4)) by
A17,
TOPREAL1: 5,
TOPREAL1:def 2
.= (((
LSeg (p00,p2))
\/ ((
LSeg (p01,p2))
\/ L2))
\/ (L3
\/ L4)) by
XBOOLE_1: 4
.= ((L2
\/ (
LSeg (p01,p2)))
\/ ((L4
\/ L3)
\/ (
LSeg (p00,p2)))) by
XBOOLE_1: 4
.= ((L2
\/ (
LSeg (p01,p2)))
\/ (L4
\/ (L3
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 4
.= ((L2
\/ (
LSeg (p01,p2)))
\/ ((
LSeg (p1,p11))
\/ ((
LSeg (p1,p10))
\/ (L3
\/ (
LSeg (p00,p2)))))) by
A29,
XBOOLE_1: 4
.= ((L2
\/ (
LSeg (p01,p2)))
\/ ((
LSeg (p1,p11))
\/ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 4
.= (((
LSeg (p1,p11))
\/ (L2
\/ (
LSeg (p01,p2))))
\/ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2)))) by
XBOOLE_1: 4
.= (P1
\/ P2) by
XBOOLE_1: 4;
A31: ((
LSeg (p1,p11))
/\ (
LSeg (p1,p10)))
=
{p1} by
A3,
TOPREAL1: 8;
A32: (P1
/\ P2)
= ((((
LSeg (p1,p11))
\/ L2)
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ ((
LSeg (p01,p2))
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
\/ L2)
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ ((
LSeg (p1,p10))
\/ L3))
\/
{p2})) by
A26,
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
\/ L2)
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ ((((
LSeg (p01,p2))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p01,p2))
/\ L3))
\/
{p2})) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ (L2
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A25,
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ ((L2
/\ ((
LSeg (p1,p10))
\/ L3))
\/ (L2
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (((
LSeg (p1,p10))
\/ L3)
\/ (
LSeg (p00,p2))))
\/ (((L2
/\ (
LSeg (p1,p10)))
\/ (L3
/\ L2))
\/ (L2
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
XBOOLE_1: 23
.= (((((
LSeg (p1,p11))
/\ ((
LSeg (p1,p10))
\/ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2))))
\/ ((L2
/\ (
LSeg (p1,p10)))
\/ (L2
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
Lm2,
XBOOLE_1: 23
.= (((
{p1}
\/ ((
LSeg (p1,p11))
/\ L3))
\/ ((L2
/\ (
LSeg (p1,p10)))
\/ (L2
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A24,
A31,
XBOOLE_1: 23;
A33:
now
per cases ;
suppose
A34: p1
= p10;
then (L2
/\ (
LSeg (p1,p10)))
= (L2
/\
{p10}) by
RLTOPSP1: 70;
then (L2
/\ (
LSeg (p1,p10)))
=
{} by
Lm1,
Lm17;
hence (P1
/\ P2)
= ((
{p1}
\/ (L2
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A32,
A34,
TOPREAL1: 16;
end;
suppose
A35: p1
= p11;
then ((
LSeg (p1,p11))
/\ L3)
= (
{p11}
/\ L3) by
RLTOPSP1: 70;
then ((
LSeg (p1,p11))
/\ L3)
=
{} by
Lm1,
Lm19;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ (L2
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A32,
A35,
TOPREAL1: 18,
XBOOLE_1: 4
.= ((
{p1}
\/ (L2
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2}));
end;
suppose
A36: p1
<> p11 & p1
<> p10;
now
assume p11
in (L2
/\ (
LSeg (p1,p10)));
then
A37: p11
in (
LSeg (p10,p1)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p11
`2 )
<= (p1
`2 ) by
A37,
TOPREAL1: 4;
then 1
= (p1
`2 ) by
A13,
A15,
Lm11,
XXREAL_0: 1;
hence contradiction by
A13,
A14,
A36,
EUCLID: 53;
end;
then
A38:
{p11}
<> (L2
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
(L2
/\ (
LSeg (p1,p10)))
c=
{p11} by
A3,
Lm25,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
then
A39: (L2
/\ (
LSeg (p1,p10)))
=
{} by
A38,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p11))
/\ L3);
then
A40: p10
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then (p1
`2 )
=
0 by
A13,
A16,
A40,
Lm9,
TOPREAL1: 4;
hence contradiction by
A13,
A14,
A36,
EUCLID: 53;
end;
then
A41:
{p10}
<> ((
LSeg (p1,p11))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L3)
c= (L4
/\ L3) by
A3,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p11))
/\ L3)
=
{} by
A41,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ (L2
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p01,p2))
/\ L3)
\/
{p2})) by
A32,
A39;
end;
end;
now
per cases ;
suppose
A42: p2
= p00;
then (L2
/\ (
LSeg (p00,p2)))
= (L2
/\
{p00}) by
RLTOPSP1: 70;
then (L2
/\ (
LSeg (p00,p2)))
=
{} by
Lm1,
Lm13;
hence thesis by
A33,
A42,
ENUMSET1: 1,
TOPREAL1: 17;
end;
suppose
A43: p2
= p01;
then ((
LSeg (p01,p2))
/\ L3)
= (
{p01}
/\ L3) by
RLTOPSP1: 70;
then ((
LSeg (p01,p2))
/\ L3)
=
{} by
Lm1,
Lm14;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A33,
A43,
TOPREAL1: 15,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A44: p2
<> p01 & p2
<> p00;
now
assume p00
in ((
LSeg (p01,p2))
/\ L3);
then
A45: p00
in (
LSeg (p2,p01)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p01
`2 ) by
A30,
EUCLID: 52;
then
0
= (p2
`2 ) by
A30,
A45,
Lm5,
TOPREAL1: 4;
hence contradiction by
A30,
A44,
EUCLID: 53;
end;
then
A46:
{p00}
<> ((
LSeg (p01,p2))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p01,p2))
/\ L3)
c=
{p00} by
A17,
Lm22,
TOPREAL1: 6,
TOPREAL1: 17,
XBOOLE_1: 26;
then
A47: ((
LSeg (p01,p2))
/\ L3)
=
{} by
A46,
ZFMISC_1: 33;
now
assume p01
in (L2
/\ (
LSeg (p00,p2)));
then
A48: p01
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`2 )
<= (p2
`2 ) by
A30,
EUCLID: 52;
then (p01
`2 )
<= (p2
`2 ) by
A48,
TOPREAL1: 4;
then (p2
`2 )
= 1 by
A30,
Lm7,
XXREAL_0: 1;
hence contradiction by
A30,
A44,
EUCLID: 53;
end;
then
A49:
{p01}
<> (L2
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
(L2
/\ (
LSeg (p00,p2)))
c= (L2
/\ L1) by
A17,
Lm20,
TOPREAL1: 6,
XBOOLE_1: 26;
then (L2
/\ (
LSeg (p00,p2)))
=
{} by
A49,
TOPREAL1: 15,
ZFMISC_1: 33;
hence thesis by
A33,
A47,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A50: p2
in L2;
then
A51: ex q st q
= p2 & (q
`1 )
<= 1 & (q
`1 )
>=
0 & (q
`2 )
= 1 by
TOPREAL1: 13;
now
A52: (p01
`1 )
<= (p2
`1 ) by
A51,
EUCLID: 52;
assume
A53: p11
in ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)));
then
A54: p11
in (
LSeg (p10,p1)) by
XBOOLE_0:def 4;
p11
in (
LSeg (p01,p2)) by
A53,
XBOOLE_0:def 4;
then (p11
`1 )
<= (p2
`1 ) by
A52,
TOPREAL1: 3;
then
A55: 1
= (p2
`1 ) by
A51,
Lm10,
XXREAL_0: 1;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p11
`2 )
<= (p1
`2 ) by
A54,
TOPREAL1: 4;
then 1
= (p1
`2 ) by
A13,
A15,
Lm11,
XXREAL_0: 1;
then p1
= p11 by
A13,
A14,
EUCLID: 53
.= p2 by
A51,
A55,
EUCLID: 53;
hence contradiction by
A1;
end;
then
A56:
{p11}
<> ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
A57: L1
is_an_arc_of (p00,p01) by
Lm5,
Lm7,
TOPREAL1: 9;
L3
is_an_arc_of (p10,p00) by
Lm4,
Lm8,
TOPREAL1: 9;
then
A58: (L3
\/ L1)
is_an_arc_of (p10,p01) by
A57,
TOPREAL1: 2,
TOPREAL1: 17;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A59: ((
LSeg (p1,p11))
/\ L1)
=
{} by
A5,
XBOOLE_1: 3,
XBOOLE_1: 26;
take P1 = ((
LSeg (p1,p11))
\/ (
LSeg (p11,p2))), P2 = ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2))));
A60: ((
LSeg (p1,p11))
\/ (
LSeg (p1,p10)))
= L4 by
A3,
TOPREAL1: 5;
p01
in (
LSeg (p01,p2)) by
RLTOPSP1: 68;
then
A61: (L1
/\ (
LSeg (p01,p2)))
<>
{} by
Lm22,
XBOOLE_0:def 4;
A62: p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A63: p11
in ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2))) by
A62,
XBOOLE_0:def 4;
A64: (
LSeg (p11,p2))
c= L2 by
A50,
Lm26,
TOPREAL1: 6;
then ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2)))
c= (L4
/\ L2) by
A5,
XBOOLE_1: 27;
then
A65: ((
LSeg (p1,p11))
/\ (
LSeg (p11,p2)))
=
{p11} by
A63,
TOPREAL1: 18,
ZFMISC_1: 33;
p1
<> p11 or p11
<> p2 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A65,
TOPREAL1: 12;
A66:
{p1}
= ((
LSeg (p1,p11))
/\ (
LSeg (p1,p10))) by
A3,
TOPREAL1: 8;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A67: ((
LSeg (p11,p2))
/\ L3)
=
{} by
A64,
XBOOLE_1: 3,
XBOOLE_1: 26;
A68: (
LSeg (p2,p01))
c= L2 by
A50,
Lm23,
TOPREAL1: 6;
then
A69: (L1
/\ (
LSeg (p01,p2)))
c=
{p01} by
TOPREAL1: 15,
XBOOLE_1: 27;
A70: (L3
/\ (
LSeg (p01,p2)))
=
{} by
A68,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L3
\/ L1)
/\ (
LSeg (p01,p2)))
= ((L3
/\ (
LSeg (p01,p2)))
\/ (L1
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.=
{p01} by
A70,
A69,
A61,
ZFMISC_1: 33;
then
A71: ((L3
\/ L1)
\/ (
LSeg (p01,p2)))
is_an_arc_of (p10,p2) by
A58,
TOPREAL1: 10;
A72:
{p2}
= ((
LSeg (p11,p2))
/\ (
LSeg (p01,p2))) by
A50,
TOPREAL1: 8;
A73: ((
LSeg (p01,p2))
\/ (
LSeg (p11,p2)))
= L2 by
A50,
TOPREAL1: 5;
((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))
c= (L4
/\ L2) by
A11,
A68,
XBOOLE_1: 27;
then
A74: ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))
=
{} by
A56,
TOPREAL1: 18,
ZFMISC_1: 33;
((
LSeg (p1,p10))
/\ ((L3
\/ L1)
\/ (
LSeg (p01,p2))))
= (((
LSeg (p1,p10))
/\ (L3
\/ L1))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p01,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ L3)
\/ ((
LSeg (p10,p1))
/\ L1)) by
A74,
XBOOLE_1: 23
.=
{p10} by
A9,
A7,
A12,
TOPREAL1: 16,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A71,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p11,p2))
\/ ((
LSeg (p1,p11))
\/ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 4
.= ((
LSeg (p11,p2))
\/ (L4
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2))))) by
A60,
XBOOLE_1: 4
.= ((
LSeg (p11,p2))
\/ ((L4
\/ (L3
\/ L1))
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= ((
LSeg (p11,p2))
\/ (((L3
\/ L4)
\/ L1)
\/ (
LSeg (p01,p2)))) by
XBOOLE_1: 4
.= ((
LSeg (p11,p2))
\/ ((L3
\/ L4)
\/ (L1
\/ (
LSeg (p01,p2))))) by
XBOOLE_1: 4
.= (((L1
\/ (
LSeg (p01,p2)))
\/ (
LSeg (p11,p2)))
\/ (L3
\/ L4)) by
XBOOLE_1: 4
.=
R^2-unit_square by
A73,
TOPREAL1:def 2,
XBOOLE_1: 4;
A75: (P1
/\ P2)
= (((
LSeg (p1,p11))
/\ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p11))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p1,p11))
/\ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ (L3
\/ L1))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
A66,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ L1))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ ((
LSeg (p11,p2))
/\ ((
LSeg (p1,p10))
\/ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
\/ ((
LSeg (p11,p2))
/\ ((L3
\/ L1)
\/ (
LSeg (p01,p2)))))) by
A59,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
\/ (((
LSeg (p11,p2))
/\ (L3
\/ L1))
\/
{p2}))) by
A72,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
\/ ((((
LSeg (p11,p2))
/\ L3)
\/ ((
LSeg (p11,p2))
/\ L1))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p11))
/\ L3)
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2}))) by
A67;
A76:
now
per cases ;
suppose
A77: p1
= p10;
then
A78: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
= ((
LSeg (p11,p2))
/\
{p10}) by
RLTOPSP1: 70;
p10
in (
LSeg (p11,p2)) implies contradiction by
A64,
Lm7,
Lm9,
Lm11,
TOPREAL1: 4;
then
A79: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A78,
Lm1;
thus (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2}))) by
A75,
A77,
TOPREAL1: 16,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2})) by
A79;
end;
suppose
A80: p1
= p11;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A81: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
<>
{} by
A80,
Lm27,
XBOOLE_0:def 4;
((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
c=
{p1} by
A64,
A80,
TOPREAL1: 18,
XBOOLE_1: 27;
then
A82: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
=
{p1} by
A81,
ZFMISC_1: 33;
((
LSeg (p1,p11))
/\ L3)
= (
{p11}
/\ L3) by
A80,
RLTOPSP1: 70;
then ((
LSeg (p1,p11))
/\ L3)
=
{} by
Lm1,
Lm19;
hence (P1
/\ P2)
= ((
{p1}
\/ (
{p1}
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2})) by
A75,
A82,
XBOOLE_1: 4
.= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2}));
end;
suppose
A83: p1
<> p11 & p1
<> p10;
now
assume p11
in ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)));
then
A84: p11
in (
LSeg (p10,p1)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p11
`2 )
<= (p1
`2 ) by
A84,
TOPREAL1: 4;
then 1
= (p1
`2 ) by
A13,
A15,
Lm11,
XXREAL_0: 1;
hence contradiction by
A13,
A14,
A83,
EUCLID: 53;
end;
then
A85:
{p11}
<> ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
c=
{p11} by
A11,
A64,
TOPREAL1: 18,
XBOOLE_1: 27;
then
A86: ((
LSeg (p11,p2))
/\ (
LSeg (p1,p10)))
=
{} by
A85,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p11))
/\ L3);
then
A87: p10
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then (p1
`2 )
=
0 by
A13,
A16,
A87,
Lm9,
TOPREAL1: 4;
hence contradiction by
A13,
A14,
A83,
EUCLID: 53;
end;
then
A88:
{p10}
<> ((
LSeg (p1,p11))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L3)
c= (L4
/\ L3) by
A3,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p11))
/\ L3)
=
{} by
A88,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))))
\/ (((
LSeg (p11,p2))
/\ L1)
\/
{p2})) by
A75,
A86;
end;
end;
now
per cases ;
suppose
A89: p2
= p01;
then
A90: ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
= ((
LSeg (p1,p11))
/\
{p01}) by
RLTOPSP1: 70;
not p01
in (
LSeg (p1,p11)) by
A5,
Lm6,
Lm8,
Lm10,
TOPREAL1: 3;
then ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
=
{} by
A90,
Lm1;
hence thesis by
A76,
A89,
ENUMSET1: 1,
TOPREAL1: 15;
end;
suppose
A91: p2
= p11;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A92: ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
<>
{} by
A91,
Lm26,
XBOOLE_0:def 4;
((
LSeg (p11,p2))
/\ L1)
= (
{p11}
/\ L1) by
A91,
RLTOPSP1: 70;
then
A93: ((
LSeg (p11,p2))
/\ L1)
=
{} by
Lm1,
Lm18;
((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
c= (L4
/\ L2) by
A5,
A68,
XBOOLE_1: 27;
then ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
=
{p2} by
A91,
A92,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A76,
A93,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A94: p2
<> p11 & p2
<> p01;
now
assume p01
in ((
LSeg (p11,p2))
/\ L1);
then
A95: p01
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p11
`1 ) by
A51,
EUCLID: 52;
then (p2
`1 )
=
0 by
A51,
A95,
Lm6,
TOPREAL1: 3;
hence contradiction by
A51,
A94,
EUCLID: 53;
end;
then
A96:
{p01}
<> ((
LSeg (p11,p2))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p11,p2))
/\ L1)
c= (L2
/\ L1) by
A50,
Lm26,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A97: ((
LSeg (p11,p2))
/\ L1)
=
{} by
A96,
TOPREAL1: 15,
ZFMISC_1: 33;
now
assume p11
in ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)));
then
A98: p11
in (
LSeg (p01,p2)) by
XBOOLE_0:def 4;
(p01
`1 )
<= (p2
`1 ) by
A51,
EUCLID: 52;
then (p11
`1 )
<= (p2
`1 ) by
A98,
TOPREAL1: 3;
then 1
= (p2
`1 ) by
A51,
Lm10,
XXREAL_0: 1;
hence contradiction by
A51,
A94,
EUCLID: 53;
end;
then
A99:
{p11}
<> ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
c= (L4
/\ L2) by
A5,
A68,
XBOOLE_1: 27;
then ((
LSeg (p1,p11))
/\ (
LSeg (p01,p2)))
=
{} by
A99,
TOPREAL1: 18,
ZFMISC_1: 33;
hence thesis by
A76,
A97,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A100: p2
in L3;
then
A101: ex q st q
= p2 & (q
`1 )
<= 1 & (q
`1 )
>=
0 & (q
`2 )
=
0 by
TOPREAL1: 13;
now
A102: (p00
`1 )
<= (p2
`1 ) by
A101,
EUCLID: 52;
assume
A103: p10
in ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)));
then
A104: p10
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
p10
in (
LSeg (p00,p2)) by
A103,
XBOOLE_0:def 4;
then (p10
`1 )
<= (p2
`1 ) by
A102,
TOPREAL1: 3;
then
A105: 1
= (p2
`1 ) by
A101,
Lm8,
XXREAL_0: 1;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then
0
= (p1
`2 ) by
A13,
A16,
A104,
Lm9,
TOPREAL1: 4;
then p1
= p10 by
A13,
A14,
EUCLID: 53;
hence contradiction by
A1,
A101,
A105,
EUCLID: 53;
end;
then
A106:
{p10}
<> ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
A107: L1
is_an_arc_of (p01,p00) by
Lm5,
Lm7,
TOPREAL1: 9;
L2
is_an_arc_of (p11,p01) by
Lm6,
Lm10,
TOPREAL1: 9;
then
A108: (L2
\/ L1)
is_an_arc_of (p11,p00) by
A107,
TOPREAL1: 2,
TOPREAL1: 15;
take P1 = ((
LSeg (p1,p10))
\/ (
LSeg (p10,p2))), P2 = ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2))));
A109: ((
LSeg (p1,p10))
\/ (
LSeg (p1,p11)))
= L4 by
A3,
TOPREAL1: 5;
p00
in (
LSeg (p00,p2)) by
RLTOPSP1: 68;
then
A110: (L1
/\ (
LSeg (p00,p2)))
<>
{} by
Lm20,
XBOOLE_0:def 4;
A111: p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A112: ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
<>
{} by
A111,
XBOOLE_0:def 4;
A113: (
LSeg (p2,p10))
c= L3 by
A100,
Lm24,
TOPREAL1: 6;
then ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
c= (L4
/\ L3) by
A11,
XBOOLE_1: 27;
then
A114: ((
LSeg (p1,p10))
/\ (
LSeg (p10,p2)))
=
{p10} by
A112,
TOPREAL1: 16,
ZFMISC_1: 33;
p1
<> p10 or p2
<> p10 by
A1;
hence P1
is_an_arc_of (p1,p2) by
A114,
TOPREAL1: 12;
A115: ((
LSeg (p1,p10))
/\ (
LSeg (p1,p11)))
=
{p1} by
A3,
TOPREAL1: 8;
(L3
/\ L2)
=
{} by
TOPREAL1: 19,
XBOOLE_0:def 7;
then
A116: ((
LSeg (p10,p2))
/\ L2)
=
{} by
A113,
XBOOLE_1: 3,
XBOOLE_1: 26;
A117: (
LSeg (p2,p00))
c= L3 by
A100,
Lm21,
TOPREAL1: 6;
then
A118: (L1
/\ (
LSeg (p00,p2)))
c=
{p00} by
TOPREAL1: 17,
XBOOLE_1: 27;
A119: (L2
/\ (
LSeg (p00,p2)))
=
{} by
A117,
Lm2,
XBOOLE_1: 3,
XBOOLE_1: 26;
((L2
\/ L1)
/\ (
LSeg (p00,p2)))
= ((L2
/\ (
LSeg (p00,p2)))
\/ (L1
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.=
{p00} by
A119,
A118,
A110,
ZFMISC_1: 33;
then
A120: ((L2
\/ L1)
\/ (
LSeg (p00,p2)))
is_an_arc_of (p11,p2) by
A108,
TOPREAL1: 10;
A121: ((
LSeg (p10,p2))
/\ (
LSeg (p00,p2)))
=
{p2} by
A100,
TOPREAL1: 8;
A122: ((
LSeg (p00,p2))
\/ (
LSeg (p10,p2)))
= L3 by
A100,
TOPREAL1: 5;
((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
c= (L4
/\ L3) by
A5,
A117,
XBOOLE_1: 27;
then
A123: ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))
=
{} by
A106,
TOPREAL1: 16,
ZFMISC_1: 33;
((
LSeg (p1,p11))
/\ ((L2
\/ L1)
\/ (
LSeg (p00,p2))))
= (((
LSeg (p1,p11))
/\ (L2
\/ L1))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p00,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ L2)
\/ ((
LSeg (p1,p11))
/\ L1)) by
A123,
XBOOLE_1: 23
.=
{p11} by
A8,
A6,
A10,
TOPREAL1: 18,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A120,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((
LSeg (p10,p2))
\/ ((
LSeg (p1,p10))
\/ ((
LSeg (p1,p11))
\/ ((L1
\/ L2)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 4
.= ((
LSeg (p10,p2))
\/ (L4
\/ ((L1
\/ L2)
\/ (
LSeg (p00,p2))))) by
A109,
XBOOLE_1: 4
.= ((((L1
\/ L2)
\/ L4)
\/ (
LSeg (p00,p2)))
\/ (
LSeg (p10,p2))) by
XBOOLE_1: 4
.= (((L1
\/ L2)
\/ L4)
\/ L3) by
A122,
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A124: (P1
/\ P2)
= (((
LSeg (p1,p10))
/\ ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((((
LSeg (p1,p10))
/\ (
LSeg (p1,p11)))
\/ ((
LSeg (p1,p10))
/\ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ (L2
\/ L1))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))) by
A115,
XBOOLE_1: 23
.= ((
{p1}
\/ ((((
LSeg (p1,p10))
/\ L2)
\/ ((
LSeg (p10,p1))
/\ L1))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ ((
LSeg (p10,p2))
/\ ((
LSeg (p1,p11))
\/ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L2)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
\/ ((
LSeg (p10,p2))
/\ ((L2
\/ L1)
\/ (
LSeg (p00,p2)))))) by
A12,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L2)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
\/ (((
LSeg (p10,p2))
/\ (L2
\/ L1))
\/
{p2}))) by
A121,
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L2)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
\/ ((((
LSeg (p10,p2))
/\ L2)
\/ ((
LSeg (p10,p2))
/\ L1))
\/
{p2}))) by
XBOOLE_1: 23
.= ((
{p1}
\/ (((
LSeg (p1,p10))
/\ L2)
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2}))) by
A116;
A125:
now
per cases ;
suppose
A126: p1
= p10;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A127: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
<>
{} by
A126,
Lm25,
XBOOLE_0:def 4;
((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
c=
{p1} by
A113,
A126,
TOPREAL1: 16,
XBOOLE_1: 27;
then
A128: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
=
{p1} by
A127,
ZFMISC_1: 33;
((
LSeg (p1,p10))
/\ L2)
= (
{p10}
/\ L2) by
A126,
RLTOPSP1: 70;
then ((
LSeg (p1,p10))
/\ L2)
=
{} by
Lm1,
Lm17;
hence (P1
/\ P2)
= ((
{p1}
\/ (
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A124,
A128,
XBOOLE_1: 4
.= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2}));
end;
suppose
A129: p1
= p11;
then
A130: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
= ((
LSeg (p10,p2))
/\
{p11}) by
RLTOPSP1: 70;
not p11
in (
LSeg (p10,p2)) by
A113,
Lm5,
Lm9,
Lm11,
TOPREAL1: 4;
then ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
=
{} by
A130,
Lm1;
hence (P1
/\ P2)
= (((
{p1}
\/
{p1})
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A124,
A129,
TOPREAL1: 18,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2}));
end;
suppose
A131: p1
<> p11 & p1
<> p10;
now
assume p10
in ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)));
then
A132: p10
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then (p1
`2 )
=
0 by
A13,
A16,
A132,
Lm9,
TOPREAL1: 4;
hence contradiction by
A13,
A14,
A131,
EUCLID: 53;
end;
then
A133:
{p10}
<> ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11))) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
c=
{p10} by
A5,
A113,
TOPREAL1: 16,
XBOOLE_1: 27;
then
A134: ((
LSeg (p10,p2))
/\ (
LSeg (p1,p11)))
=
{} by
A133,
ZFMISC_1: 33;
now
assume p11
in ((
LSeg (p1,p10))
/\ L2);
then
A135: p11
in (
LSeg (p10,p1)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p11
`2 )
<= (p1
`2 ) by
A135,
TOPREAL1: 4;
then (p1
`2 )
= 1 by
A13,
A15,
Lm11,
XXREAL_0: 1;
hence contradiction by
A13,
A14,
A131,
EUCLID: 53;
end;
then
A136: ((
LSeg (p1,p10))
/\ L2)
<>
{p11} by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ L2)
c= (L4
/\ L2) by
A3,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
then ((
LSeg (p1,p10))
/\ L2)
=
{} by
A136,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
= ((
{p1}
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))))
\/ (((
LSeg (p10,p2))
/\ L1)
\/
{p2})) by
A124,
A134;
end;
end;
now
per cases ;
suppose
A137: p2
= p00;
then
A138: ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
= ((
LSeg (p1,p10))
/\
{p00}) by
RLTOPSP1: 70;
not p00
in (
LSeg (p1,p10)) by
A11,
Lm4,
Lm8,
Lm10,
TOPREAL1: 3;
then ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
=
{} by
A138,
Lm1;
hence thesis by
A125,
A137,
ENUMSET1: 1,
TOPREAL1: 17;
end;
suppose
A139: p2
= p10;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A140: ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
<>
{} by
A139,
Lm24,
XBOOLE_0:def 4;
((
LSeg (p10,p2))
/\ L1)
= (
{p10}
/\ L1) by
A139,
RLTOPSP1: 70;
then
A141: ((
LSeg (p10,p2))
/\ L1)
=
{} by
Lm1,
Lm16;
((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
c= (L4
/\ L3) by
A11,
A117,
XBOOLE_1: 27;
then ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
=
{p2} by
A139,
A140,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (
{p2}
\/
{p2})) by
A125,
A141,
XBOOLE_1: 4
.=
{p1, p2} by
ENUMSET1: 1;
end;
suppose
A142: p2
<> p10 & p2
<> p00;
now
assume p00
in ((
LSeg (p10,p2))
/\ L1);
then
A143: p00
in (
LSeg (p2,p10)) by
XBOOLE_0:def 4;
(p2
`1 )
<= (p10
`1 ) by
A101,
EUCLID: 52;
then (p2
`1 )
=
0 by
A101,
A143,
Lm4,
TOPREAL1: 3;
hence contradiction by
A101,
A142,
EUCLID: 53;
end;
then
A144:
{p00}
<> ((
LSeg (p10,p2))
/\ L1) by
ZFMISC_1: 31;
((
LSeg (p10,p2))
/\ L1)
c= (L3
/\ L1) by
A100,
Lm24,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A145: ((
LSeg (p10,p2))
/\ L1)
=
{} by
A144,
TOPREAL1: 17,
ZFMISC_1: 33;
now
assume p10
in ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)));
then
A146: p10
in (
LSeg (p00,p2)) by
XBOOLE_0:def 4;
(p00
`1 )
<= (p2
`1 ) by
A101,
EUCLID: 52;
then (p10
`1 )
<= (p2
`1 ) by
A146,
TOPREAL1: 3;
then (p2
`1 )
= 1 by
A101,
Lm8,
XXREAL_0: 1;
hence contradiction by
A101,
A142,
EUCLID: 53;
end;
then
A147:
{p10}
<> ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2))) by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
c= (L4
/\ L3) by
A11,
A117,
XBOOLE_1: 27;
then ((
LSeg (p1,p10))
/\ (
LSeg (p00,p2)))
=
{} by
A147,
TOPREAL1: 16,
ZFMISC_1: 33;
hence thesis by
A125,
A145,
ENUMSET1: 1;
end;
end;
hence thesis;
end;
suppose
A148: p2
in L4;
A149: p
=
|[(p
`1 ), (p
`2 )]| by
EUCLID: 53;
A150: (
LSeg (p1,p2))
c= L4 by
A3,
A148,
TOPREAL1: 6;
consider q such that
A151: q
= p2 and
A152: (q
`1 )
= 1 and
A153: (q
`2 )
<= 1 and
A154: (q
`2 )
>=
0 by
A148,
TOPREAL1: 13;
A155: q
=
|[(q
`1 ), (q
`2 )]| by
EUCLID: 53;
now
per cases by
A1,
A13,
A14,
A151,
A152,
A149,
A155,
XXREAL_0: 1;
suppose
A156: (p
`2 )
< (q
`2 );
A157: ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)))
c=
{p2}
proof
let a be
object;
assume
A158: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A159: p
in (
LSeg (p2,p11)) by
A158,
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A151,
A153,
EUCLID: 52;
then
A160: (p2
`2 )
<= (p
`2 ) by
A159,
TOPREAL1: 4;
A161: p
in (
LSeg (p1,p2)) by
A158,
XBOOLE_0:def 4;
then
A162: (p1
`1 )
<= (p
`1 ) by
A13,
A14,
A151,
A152,
TOPREAL1: 3;
(p
`2 )
<= (p2
`2 ) by
A13,
A151,
A156,
A161,
TOPREAL1: 4;
then
A163: (p2
`2 )
= (p
`2 ) by
A160,
XXREAL_0: 1;
(p
`1 )
<= (p2
`1 ) by
A13,
A14,
A151,
A152,
A161,
TOPREAL1: 3;
then (p
`1 )
= 1 by
A13,
A14,
A151,
A152,
A162,
XXREAL_0: 1;
then p
=
|[1, (p2
`2 )]| by
A163,
EUCLID: 53
.= p2 by
A151,
A152,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
p10
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
then
A164: ((
LSeg (p1,p10))
/\ L3)
<>
{} by
Lm24,
XBOOLE_0:def 4;
A165:
now
set a = the
Element of ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)));
assume
A166: ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A167: p
in (
LSeg (p10,p1)) by
A166,
XBOOLE_0:def 4;
A168: p
in (
LSeg (p2,p11)) by
A166,
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A151,
A153,
EUCLID: 52;
then
A169: (p2
`2 )
<= (p
`2 ) by
A168,
TOPREAL1: 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p
`2 )
<= (p1
`2 ) by
A167,
TOPREAL1: 4;
hence contradiction by
A13,
A151,
A156,
A169,
XXREAL_0: 2;
end;
A170: ((L3
\/ L1)
/\ L2)
= ((L3
/\ L2)
\/ (L1
/\ L2)) by
XBOOLE_1: 23
.=
{p01} by
Lm2,
TOPREAL1: 15;
(L3
\/ L1)
is_an_arc_of (p10,p01) by
Lm4,
Lm8,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 17;
then
A171: ((L3
\/ L1)
\/ L2)
is_an_arc_of (p10,p11) by
A170,
TOPREAL1: 10;
now
assume p11
in ((
LSeg (p1,p10))
/\ L2);
then
A172: p11
in (
LSeg (p10,p1)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then (p11
`2 )
<= (p1
`2 ) by
A172,
TOPREAL1: 4;
hence contradiction by
A13,
A15,
A153,
A156,
Lm11,
XXREAL_0: 1;
end;
then
A173:
{p11}
<> ((
LSeg (p1,p10))
/\ L2) by
ZFMISC_1: 31;
((
LSeg (p1,p10))
/\ L2)
c= (L4
/\ L2) by
A3,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A174: ((
LSeg (p1,p10))
/\ L2)
=
{} by
A173,
TOPREAL1: 18,
ZFMISC_1: 33;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A175: ((
LSeg (p1,p10))
/\ L1)
=
{} by
A11,
XBOOLE_1: 3,
XBOOLE_1: 26;
now
assume p10
in (L3
/\ (
LSeg (p11,p2)));
then
A176: p10
in (
LSeg (p2,p11)) by
XBOOLE_0:def 4;
(p2
`2 )
<= (p11
`2 ) by
A151,
A153,
EUCLID: 52;
hence contradiction by
A16,
A151,
A156,
A176,
Lm9,
TOPREAL1: 4;
end;
then
A177:
{p10}
<> (L3
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
(L3
/\ (
LSeg (p11,p2)))
c=
{p10} by
A148,
Lm27,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then
A178: (L3
/\ (
LSeg (p11,p2)))
=
{} by
A177,
ZFMISC_1: 33;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A179: ((
LSeg (p1,p2))
/\ L1)
=
{} by
A150,
XBOOLE_1: 3,
XBOOLE_1: 26;
A180: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)))
c=
{p1}
proof
let a be
object;
assume
A181: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A182: p
in (
LSeg (p10,p1)) by
A181,
XBOOLE_0:def 4;
(p10
`2 )
<= (p1
`2 ) by
A13,
A16,
EUCLID: 52;
then
A183: (p
`2 )
<= (p1
`2 ) by
A182,
TOPREAL1: 4;
A184: p
in (
LSeg (p1,p2)) by
A181,
XBOOLE_0:def 4;
then
A185: (p1
`1 )
<= (p
`1 ) by
A13,
A14,
A151,
A152,
TOPREAL1: 3;
(p1
`2 )
<= (p
`2 ) by
A13,
A151,
A156,
A184,
TOPREAL1: 4;
then
A186: (p1
`2 )
= (p
`2 ) by
A183,
XXREAL_0: 1;
(p
`1 )
<= (p2
`1 ) by
A13,
A14,
A151,
A152,
A184,
TOPREAL1: 3;
then (p
`1 )
= 1 by
A13,
A14,
A151,
A152,
A185,
XXREAL_0: 1;
then p
=
|[1, (p1
`2 )]| by
A186,
EUCLID: 53
.= p1 by
A13,
A14,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A187: ((
LSeg (p1,p10))
/\ L3)
c= (L4
/\ L3) by
A3,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
p11
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
then
A188: (L2
/\ (
LSeg (p11,p2)))
<>
{} by
Lm26,
XBOOLE_0:def 4;
(L2
/\ (
LSeg (p11,p2)))
c=
{p11} by
A148,
Lm27,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
then
A189: (L2
/\ (
LSeg (p11,p2)))
=
{p11} by
A188,
ZFMISC_1: 33;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p10))
\/ (((L3
\/ L1)
\/ L2)
\/ (
LSeg (p11,p2))));
A190: p1
in (
LSeg (p1,p10)) by
RLTOPSP1: 68;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A191: (L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
(L1
/\ (
LSeg (p11,p2)))
c= (L1
/\ L4) by
A148,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A192: (L1
/\ (
LSeg (p11,p2)))
=
{} by
A191,
XBOOLE_1: 3;
(((L3
\/ L1)
\/ L2)
/\ (
LSeg (p11,p2)))
= (((L3
\/ L1)
/\ (
LSeg (p11,p2)))
\/ (L2
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((L3
/\ (
LSeg (p11,p2)))
\/ (L1
/\ (
LSeg (p11,p2))))
\/
{p11}) by
A189,
XBOOLE_1: 23
.=
{p11} by
A178,
A192;
then
A193: (((L3
\/ L1)
\/ L2)
\/ (
LSeg (p11,p2)))
is_an_arc_of (p10,p2) by
A171,
TOPREAL1: 10;
((
LSeg (p1,p10))
/\ (((L3
\/ L1)
\/ L2)
\/ (
LSeg (p11,p2))))
= (((
LSeg (p1,p10))
/\ ((L3
\/ L1)
\/ L2))
\/ ((
LSeg (p1,p10))
/\ (
LSeg (p11,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ (L3
\/ L1))
\/ ((
LSeg (p1,p10))
/\ L2)) by
A165,
XBOOLE_1: 23
.= (((
LSeg (p1,p10))
/\ L3)
\/ ((
LSeg (p1,p10))
/\ L1)) by
A174,
XBOOLE_1: 23
.=
{p10} by
A187,
A164,
A175,
TOPREAL1: 16,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A193,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((((L3
\/ L1)
\/ L2)
\/ (
LSeg (p11,p2)))
\/ ((
LSeg (p1,p10))
\/ (
LSeg (p1,p2)))) by
XBOOLE_1: 4
.= (((L3
\/ L1)
\/ L2)
\/ (((
LSeg (p10,p1))
\/ (
LSeg (p1,p2)))
\/ (
LSeg (p2,p11)))) by
XBOOLE_1: 4
.= (((L3
\/ L1)
\/ L2)
\/ L4) by
A3,
A148,
TOPREAL1: 7
.= ((L3
\/ (L1
\/ L2))
\/ L4) by
XBOOLE_1: 4
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A194: p2
in (
LSeg (p11,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2))) by
A194,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2))) by
ZFMISC_1: 31;
then
A195: ((
LSeg (p1,p2))
/\ (
LSeg (p11,p2)))
=
{p2} by
A157,
XBOOLE_0:def 10;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10))) by
A190,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10))) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ (
LSeg (p1,p10)))
=
{p1} by
A180,
XBOOLE_0:def 10;
then
A196: (P1
/\ P2)
= (
{p1}
\/ ((
LSeg (p1,p2))
/\ (((L3
\/ L1)
\/ L2)
\/ (
LSeg (p11,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L3
\/ L1)
\/ L2))
\/
{p2})) by
A195,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L3
\/ L1))
\/ ((
LSeg (p1,p2))
/\ L2))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L3)
\/ ((
LSeg (p1,p2))
/\ L1))
\/ ((
LSeg (p1,p2))
/\ L2))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L3)
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2}))) by
A179,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L3))
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
XBOOLE_1: 4;
A197: ((
LSeg (p1,p2))
/\ L3)
c= (L4
/\ L3) by
A3,
A148,
TOPREAL1: 6,
XBOOLE_1: 26;
A198:
now
per cases ;
suppose
A199: p1
= p10;
then p10
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L3)
<>
{} by
Lm24,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L3)
=
{p1} by
A197,
A199,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
A196;
end;
suppose
A200: p1
<> p10;
now
assume p10
in ((
LSeg (p1,p2))
/\ L3);
then p10
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p1
`2 )
=
0 by
A13,
A16,
A151,
A156,
Lm9,
TOPREAL1: 4;
hence contradiction by
A13,
A14,
A200,
EUCLID: 53;
end;
then
{p10}
<> ((
LSeg (p1,p2))
/\ L3) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L3)
=
{} by
A197,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L2)
\/
{p2})) by
A196;
end;
end;
A201: ((
LSeg (p1,p2))
/\ L2)
c= (L4
/\ L2) by
A3,
A148,
TOPREAL1: 6,
XBOOLE_1: 26;
now
per cases ;
suppose
A202: p2
= p11;
then p11
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L2)
<>
{} by
Lm26,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L2)
=
{p2} by
A201,
A202,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A198,
ENUMSET1: 1;
end;
suppose
A203: p2
<> p11;
now
assume p11
in ((
LSeg (p1,p2))
/\ L2);
then p11
in (
LSeg (p1,p2)) by
XBOOLE_0:def 4;
then (p11
`2 )
<= (p2
`2 ) by
A13,
A151,
A156,
TOPREAL1: 4;
then (p2
`2 )
= 1 by
A151,
A153,
Lm11,
XXREAL_0: 1;
hence contradiction by
A151,
A152,
A203,
EUCLID: 53;
end;
then
{p11}
<> ((
LSeg (p1,p2))
/\ L2) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L2)
=
{} by
A201,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A198,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
suppose
A204: (q
`2 )
< (p
`2 );
A205: ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)))
c=
{p2}
proof
let a be
object;
assume
A206: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A207: p
in (
LSeg (p10,p2)) by
A206,
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A151,
A154,
EUCLID: 52;
then
A208: (p
`2 )
<= (p2
`2 ) by
A207,
TOPREAL1: 4;
A209: p
in (
LSeg (p2,p1)) by
A206,
XBOOLE_0:def 4;
then
A210: (p2
`1 )
<= (p
`1 ) by
A13,
A14,
A151,
A152,
TOPREAL1: 3;
(p2
`2 )
<= (p
`2 ) by
A13,
A151,
A204,
A209,
TOPREAL1: 4;
then
A211: (p2
`2 )
= (p
`2 ) by
A208,
XXREAL_0: 1;
(p
`1 )
<= (p1
`1 ) by
A13,
A14,
A151,
A152,
A209,
TOPREAL1: 3;
then (p
`1 )
= 1 by
A13,
A14,
A151,
A152,
A210,
XXREAL_0: 1;
then p
=
|[1, (p2
`2 )]| by
A211,
EUCLID: 53
.= p2 by
A151,
A152,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
p11
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
then
A212: ((
LSeg (p1,p11))
/\ L2)
<>
{} by
Lm26,
XBOOLE_0:def 4;
A213:
now
set a = the
Element of ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)));
assume
A214: ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))
<>
{} ;
then
reconsider p = a as
Point of (
TOP-REAL 2) by
TARSKI:def 3;
A215: p
in (
LSeg (p1,p11)) by
A214,
XBOOLE_0:def 4;
A216: p
in (
LSeg (p10,p2)) by
A214,
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A151,
A154,
EUCLID: 52;
then
A217: (p
`2 )
<= (p2
`2 ) by
A216,
TOPREAL1: 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then (p1
`2 )
<= (p
`2 ) by
A215,
TOPREAL1: 4;
hence contradiction by
A13,
A151,
A204,
A217,
XXREAL_0: 2;
end;
A218: ((L2
\/ L1)
/\ L3)
= ((L3
/\ L2)
\/ (L1
/\ L3)) by
XBOOLE_1: 23
.=
{p00} by
Lm2,
TOPREAL1: 17;
(L2
\/ L1)
is_an_arc_of (p11,p00) by
Lm6,
Lm10,
TOPREAL1: 9,
TOPREAL1: 10,
TOPREAL1: 15;
then
A219: ((L2
\/ L1)
\/ L3)
is_an_arc_of (p11,p10) by
A218,
TOPREAL1: 10;
now
assume p11
in (L2
/\ (
LSeg (p10,p2)));
then
A220: p11
in (
LSeg (p10,p2)) by
XBOOLE_0:def 4;
(p10
`2 )
<= (p2
`2 ) by
A151,
A154,
EUCLID: 52;
then (p11
`2 )
<= (p2
`2 ) by
A220,
TOPREAL1: 4;
hence contradiction by
A15,
A151,
A153,
A204,
Lm11,
XXREAL_0: 1;
end;
then
A221:
{p11}
<> (L2
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
(L2
/\ (
LSeg (p10,p2)))
c=
{p11} by
A148,
Lm25,
TOPREAL1: 6,
TOPREAL1: 18,
XBOOLE_1: 26;
then
A222: (L2
/\ (
LSeg (p10,p2)))
=
{} by
A221,
ZFMISC_1: 33;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A223: ((
LSeg (p1,p11))
/\ L1)
=
{} by
A5,
XBOOLE_1: 3,
XBOOLE_1: 26;
now
assume p10
in ((
LSeg (p1,p11))
/\ L3);
then
A224: p10
in (
LSeg (p1,p11)) by
XBOOLE_0:def 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
hence contradiction by
A13,
A154,
A204,
A224,
Lm9,
TOPREAL1: 4;
end;
then
A225:
{p10}
<> ((
LSeg (p1,p11))
/\ L3) by
ZFMISC_1: 31;
((
LSeg (p1,p11))
/\ L3)
c= (L4
/\ L3) by
A3,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A226: ((
LSeg (p1,p11))
/\ L3)
=
{} by
A225,
TOPREAL1: 16,
ZFMISC_1: 33;
(L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
then
A227: ((
LSeg (p1,p2))
/\ L1)
=
{} by
A150,
XBOOLE_1: 3,
XBOOLE_1: 26;
A228: ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)))
c=
{p1}
proof
let a be
object;
assume
A229: a
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)));
then
reconsider p = a as
Point of (
TOP-REAL 2);
A230: p
in (
LSeg (p1,p11)) by
A229,
XBOOLE_0:def 4;
(p1
`2 )
<= (p11
`2 ) by
A13,
A15,
EUCLID: 52;
then
A231: (p1
`2 )
<= (p
`2 ) by
A230,
TOPREAL1: 4;
A232: p
in (
LSeg (p2,p1)) by
A229,
XBOOLE_0:def 4;
then
A233: (p2
`1 )
<= (p
`1 ) by
A13,
A14,
A151,
A152,
TOPREAL1: 3;
(p
`2 )
<= (p1
`2 ) by
A13,
A151,
A204,
A232,
TOPREAL1: 4;
then
A234: (p1
`2 )
= (p
`2 ) by
A231,
XXREAL_0: 1;
(p
`1 )
<= (p1
`1 ) by
A13,
A14,
A151,
A152,
A232,
TOPREAL1: 3;
then (p
`1 )
= 1 by
A13,
A14,
A151,
A152,
A233,
XXREAL_0: 1;
then p
=
|[1, (p1
`2 )]| by
A234,
EUCLID: 53
.= p1 by
A13,
A14,
EUCLID: 53;
hence thesis by
TARSKI:def 1;
end;
A235: ((
LSeg (p1,p11))
/\ L2)
c= (L4
/\ L2) by
A3,
Lm27,
TOPREAL1: 6,
XBOOLE_1: 26;
p10
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
then
A236: (L3
/\ (
LSeg (p10,p2)))
<>
{} by
Lm24,
XBOOLE_0:def 4;
(L3
/\ (
LSeg (p10,p2)))
c=
{p10} by
A148,
Lm25,
TOPREAL1: 6,
TOPREAL1: 16,
XBOOLE_1: 26;
then
A237: (L3
/\ (
LSeg (p10,p2)))
=
{p10} by
A236,
ZFMISC_1: 33;
take P1 = (
LSeg (p1,p2)), P2 = ((
LSeg (p1,p11))
\/ (((L2
\/ L1)
\/ L3)
\/ (
LSeg (p10,p2))));
A238: p1
in (
LSeg (p1,p11)) by
RLTOPSP1: 68;
thus P1
is_an_arc_of (p1,p2) by
A1,
TOPREAL1: 9;
A239: (L1
/\ L4)
=
{} by
TOPREAL1: 20,
XBOOLE_0:def 7;
(L1
/\ (
LSeg (p10,p2)))
c= (L1
/\ L4) by
A148,
Lm25,
TOPREAL1: 6,
XBOOLE_1: 26;
then
A240: (L1
/\ (
LSeg (p10,p2)))
=
{} by
A239,
XBOOLE_1: 3;
(((L2
\/ L1)
\/ L3)
/\ (
LSeg (p10,p2)))
= (((L2
\/ L1)
/\ (
LSeg (p10,p2)))
\/ (L3
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((L2
/\ (
LSeg (p10,p2)))
\/ (L1
/\ (
LSeg (p10,p2))))
\/
{p10}) by
A237,
XBOOLE_1: 23
.=
{p10} by
A222,
A240;
then
A241: (((L2
\/ L1)
\/ L3)
\/ (
LSeg (p10,p2)))
is_an_arc_of (p11,p2) by
A219,
TOPREAL1: 10;
((
LSeg (p1,p11))
/\ (((L2
\/ L1)
\/ L3)
\/ (
LSeg (p10,p2))))
= (((
LSeg (p1,p11))
/\ ((L2
\/ L1)
\/ L3))
\/ ((
LSeg (p1,p11))
/\ (
LSeg (p10,p2)))) by
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ (L2
\/ L1))
\/ ((
LSeg (p1,p11))
/\ L3)) by
A213,
XBOOLE_1: 23
.= (((
LSeg (p1,p11))
/\ L2)
\/ ((
LSeg (p1,p11))
/\ L1)) by
A226,
XBOOLE_1: 23
.=
{p11} by
A235,
A212,
A223,
TOPREAL1: 18,
ZFMISC_1: 33;
hence P2
is_an_arc_of (p1,p2) by
A241,
TOPREAL1: 11;
thus (P1
\/ P2)
= ((((L2
\/ L1)
\/ L3)
\/ (
LSeg (p10,p2)))
\/ ((
LSeg (p1,p11))
\/ (
LSeg (p1,p2)))) by
XBOOLE_1: 4
.= (((L2
\/ L1)
\/ L3)
\/ ((
LSeg (p10,p2))
\/ ((
LSeg (p1,p2))
\/ (
LSeg (p1,p11))))) by
XBOOLE_1: 4
.= (((L1
\/ L2)
\/ L3)
\/ L4) by
A3,
A148,
TOPREAL1: 7
.=
R^2-unit_square by
TOPREAL1:def 2,
XBOOLE_1: 4;
A242: p2
in (
LSeg (p10,p2)) by
RLTOPSP1: 68;
p2
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p2
in ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2))) by
A242,
XBOOLE_0:def 4;
then
{p2}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2))) by
ZFMISC_1: 31;
then
A243: ((
LSeg (p1,p2))
/\ (
LSeg (p10,p2)))
=
{p2} by
A205,
XBOOLE_0:def 10;
p1
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then p1
in ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11))) by
A238,
XBOOLE_0:def 4;
then
{p1}
c= ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11))) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ (
LSeg (p1,p11)))
=
{p1} by
A228,
XBOOLE_0:def 10;
then
A244: (P1
/\ P2)
= (
{p1}
\/ ((
LSeg (p1,p2))
/\ (((L2
\/ L1)
\/ L3)
\/ (
LSeg (p10,p2))))) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ ((L2
\/ L1)
\/ L3))
\/
{p2})) by
A243,
XBOOLE_1: 23
.= (
{p1}
\/ ((((
LSeg (p1,p2))
/\ (L2
\/ L1))
\/ ((
LSeg (p1,p2))
/\ L3))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((((
LSeg (p1,p2))
/\ L2)
\/ ((
LSeg (p1,p2))
/\ L1))
\/ ((
LSeg (p1,p2))
/\ L3))
\/
{p2})) by
XBOOLE_1: 23
.= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L2)
\/ (((
LSeg (p1,p2))
/\ L3)
\/
{p2}))) by
A227,
XBOOLE_1: 4
.= ((
{p1}
\/ ((
LSeg (p1,p2))
/\ L2))
\/ (((
LSeg (p1,p2))
/\ L3)
\/
{p2})) by
XBOOLE_1: 4;
A245: ((
LSeg (p1,p2))
/\ L2)
c= (L4
/\ L2) by
A3,
A148,
TOPREAL1: 6,
XBOOLE_1: 26;
A246:
now
per cases ;
suppose
A247: p1
= p11;
then p11
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L2)
<>
{} by
Lm26,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L2)
=
{p1} by
A245,
A247,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L3)
\/
{p2})) by
A244;
end;
suppose
A248: p1
<> p11;
now
assume p11
in ((
LSeg (p1,p2))
/\ L2);
then p11
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p11
`2 )
<= (p1
`2 ) by
A13,
A151,
A204,
TOPREAL1: 4;
then (p1
`2 )
= 1 by
A13,
A15,
Lm11,
XXREAL_0: 1;
hence contradiction by
A13,
A14,
A248,
EUCLID: 53;
end;
then
{p11}
<> ((
LSeg (p1,p2))
/\ L2) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L2)
=
{} by
A245,
TOPREAL1: 18,
ZFMISC_1: 33;
hence (P1
/\ P2)
= (
{p1}
\/ (((
LSeg (p1,p2))
/\ L3)
\/
{p2})) by
A244;
end;
end;
A249: ((
LSeg (p1,p2))
/\ L3)
c= (L4
/\ L3) by
A3,
A148,
TOPREAL1: 6,
XBOOLE_1: 26;
now
per cases ;
suppose
A250: p2
= p10;
then p10
in (
LSeg (p1,p2)) by
RLTOPSP1: 68;
then ((
LSeg (p1,p2))
/\ L3)
<>
{} by
Lm24,
XBOOLE_0:def 4;
then ((
LSeg (p1,p2))
/\ L3)
=
{p2} by
A249,
A250,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A246,
ENUMSET1: 1;
end;
suppose
A251: p2
<> p10;
now
assume p10
in ((
LSeg (p1,p2))
/\ L3);
then p10
in (
LSeg (p2,p1)) by
XBOOLE_0:def 4;
then (p2
`2 )
=
0 by
A13,
A151,
A154,
A204,
Lm9,
TOPREAL1: 4;
hence contradiction by
A151,
A152,
A251,
EUCLID: 53;
end;
then
{p10}
<> ((
LSeg (p1,p2))
/\ L3) by
ZFMISC_1: 31;
then ((
LSeg (p1,p2))
/\ L3)
=
{} by
A249,
TOPREAL1: 16,
ZFMISC_1: 33;
hence (P1
/\ P2)
=
{p1, p2} by
A246,
ENUMSET1: 1;
end;
end;
hence (P1
/\ P2)
=
{p1, p2};
end;
end;
hence thesis;
end;
end;
theorem ::
TOPREAL2:1
Th1: p1
<> p2 & p1
in
R^2-unit_square & p2
in
R^2-unit_square implies ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) &
R^2-unit_square
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume that
A1: p1
<> p2 and
A2: p1
in
R^2-unit_square and
A3: p2
in
R^2-unit_square ;
A4: p1
in (L1
\/ L2) or p1
in (L3
\/ L4) by
A2,
TOPREAL1:def 2,
XBOOLE_0:def 3;
per cases by
A4,
XBOOLE_0:def 3;
suppose p1
in L1;
hence thesis by
A1,
A3,
Lm30;
end;
suppose p1
in L2;
hence thesis by
A1,
A3,
Lm31;
end;
suppose p1
in L3;
hence thesis by
A1,
A3,
Lm32;
end;
suppose p1
in L4;
hence thesis by
A1,
A3,
Lm33;
end;
end;
theorem ::
TOPREAL2:2
Th2:
R^2-unit_square is
compact
proof
A1:
I[01] is
compact by
HEINE: 4,
TOPMETR: 20;
consider P1,P2 be non
empty
Subset of (
TOP-REAL 2) such that
A2: P1 is
being_S-P_arc and
A3: P2 is
being_S-P_arc and
A4:
R^2-unit_square
= (P1
\/ P2) by
TOPREAL1: 27;
consider f be
Function of
I[01] , ((
TOP-REAL 2)
| P1) such that
A5: f is
being_homeomorphism by
A2,
TOPREAL1: 29;
A6: (
rng f)
= (
[#] ((
TOP-REAL 2)
| P1)) by
A5;
consider f0 be
Function of
I[01] , ((
TOP-REAL 2)
| P2) such that
A7: f0 is
being_homeomorphism by
A3,
TOPREAL1: 29;
A8: (
rng f0)
= (
[#] ((
TOP-REAL 2)
| P2)) by
A7;
reconsider P2 as non
empty
Subset of (
TOP-REAL 2);
f0 is
continuous by
A7;
then ((
TOP-REAL 2)
| P2) is
compact by
A1,
A8,
COMPTS_1: 14;
then
A9: P2 is
compact by
COMPTS_1: 3;
reconsider P1 as non
empty
Subset of (
TOP-REAL 2);
f is
continuous by
A5;
then ((
TOP-REAL 2)
| P1) is
compact by
A1,
A6,
COMPTS_1: 14;
then P1 is
compact by
COMPTS_1: 3;
hence thesis by
A4,
A9,
COMPTS_1: 10;
end;
theorem ::
TOPREAL2:3
Th3: for Q,P be non
empty
Subset of (
TOP-REAL 2) holds for f be
Function of ((
TOP-REAL 2)
| Q), ((
TOP-REAL 2)
| P) st f is
being_homeomorphism & Q
is_an_arc_of (q1,q2) holds for p1, p2 st p1
= (f
. q1) & p2
= (f
. q2) holds P
is_an_arc_of (p1,p2)
proof
let Q,P be non
empty
Subset of (
TOP-REAL 2);
let f be
Function of ((
TOP-REAL 2)
| Q), ((
TOP-REAL 2)
| P);
assume that
A1: f is
being_homeomorphism and
A2: Q
is_an_arc_of (q1,q2);
let p1, p2 such that
A3: p1
= (f
. q1) and
A4: p2
= (f
. q2);
reconsider f as
Function of ((
TOP-REAL 2)
| Q), ((
TOP-REAL 2)
| P);
consider f1 be
Function of
I[01] , ((
TOP-REAL 2)
| Q) such that
A5: f1 is
being_homeomorphism and
A6: (f1
.
0 )
= q1 and
A7: (f1
. 1)
= q2 by
A2,
TOPREAL1:def 1;
set g1 = (f
* f1);
A8: (
dom f1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
then
0
in (
dom f1) by
BORSUK_1: 40,
XXREAL_1: 1;
then
A9: (g1
.
0 )
= p1 by
A3,
A6,
FUNCT_1: 13;
1
in (
dom f1) by
A8,
BORSUK_1: 40,
XXREAL_1: 1;
then
A10: (g1
. 1)
= p2 by
A4,
A7,
FUNCT_1: 13;
g1 is
being_homeomorphism by
A1,
A5,
TOPS_2: 57;
hence thesis by
A9,
A10,
TOPREAL1:def 1;
end;
definition
let P be
Subset of (
TOP-REAL 2);
::
TOPREAL2:def1
attr P is
being_simple_closed_curve means ex f be
Function of ((
TOP-REAL 2)
|
R^2-unit_square ), ((
TOP-REAL 2)
| P) st f is
being_homeomorphism;
end
registration
cluster
R^2-unit_square ->
being_simple_closed_curve;
coherence
proof
set T = ((
TOP-REAL 2)
|
R^2-unit_square );
take f = (
id T);
thus (
dom f)
= (
[#] T) by
FUNCT_2:def 1;
thus (
rng f)
= (
[#] T) by
RELAT_1: 45;
then f is
onto
one-to-one by
FUNCT_2:def 3;
then
A1: (f
" )
= (f qua
Function
" ) by
TOPS_2:def 4
.= f by
FUNCT_1: 45;
thus f is
one-to-one;
thus f is
continuous by
FUNCT_2: 94;
hence thesis by
A1;
end;
end
registration
cluster
being_simple_closed_curve non
empty for
Subset of (
TOP-REAL 2);
existence
proof
take
R^2-unit_square ;
thus thesis;
end;
end
definition
mode
Simple_closed_curve is
being_simple_closed_curve
Subset of (
TOP-REAL 2);
end
theorem ::
TOPREAL2:4
Th4: for P be non
empty
Subset of (
TOP-REAL 2) st P is
being_simple_closed_curve holds ex p1, p2 st p1
<> p2 & p1
in P & p2
in P
proof
reconsider RS =
R^2-unit_square as non
empty
Subset of (
TOP-REAL 2);
let P be non
empty
Subset of (
TOP-REAL 2);
A1: (p00
`1 )
=
0 by
EUCLID: 52;
A2: (
[#] ((
TOP-REAL 2)
| P))
c= (
[#] (
TOP-REAL 2)) by
PRE_TOPC:def 4;
A3: (p11
`1 )
= 1 by
EUCLID: 52;
assume P is
being_simple_closed_curve;
then
consider f be
Function of ((
TOP-REAL 2)
|
R^2-unit_square ), ((
TOP-REAL 2)
| P) such that
A4: f is
being_homeomorphism;
A5: (
rng f)
= (
[#] ((
TOP-REAL 2)
| P)) by
A4
.= P by
PRE_TOPC:def 5;
reconsider f as
Function of ((
TOP-REAL 2)
| RS), ((
TOP-REAL 2)
| P);
A6: (
dom f)
= (
[#] ((
TOP-REAL 2)
| RS)) by
FUNCT_2:def 1
.=
R^2-unit_square by
PRE_TOPC:def 5;
set p1 = (f
. p00), p2 = (f
. p11);
(p00
`2 )
=
0 by
EUCLID: 52;
then
A7: p00
in (
dom f) by
A1,
A6,
TOPREAL1: 14;
then
A8: p1
in (
rng f) by
FUNCT_1:def 3;
(p11
`2 )
= 1 by
EUCLID: 52;
then
A9: p11
in (
dom f) by
A3,
A6,
TOPREAL1: 14;
then
A10: p2
in (
rng f) by
FUNCT_1:def 3;
reconsider p1, p2 as
Point of (
TOP-REAL 2) by
A2,
A8,
A10;
take p1, p2;
f is
one-to-one by
A4;
hence p1
<> p2 by
A1,
A3,
A7,
A9,
FUNCT_1:def 4;
thus thesis by
A5,
A7,
A9,
FUNCT_1:def 3;
end;
Lm34: for P,P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2} holds P is
being_simple_closed_curve
proof
reconsider RS =
R^2-unit_square as non
empty
Subset of (
TOP-REAL 2);
let P,P1,P2 be non
empty
Subset of (
TOP-REAL 2) such that
A1: P1
is_an_arc_of (p1,p2) and
A2: P2
is_an_arc_of (p1,p2) and
A3: P
= (P1
\/ P2) and
A4: (P1
/\ P2)
=
{p1, p2};
reconsider P9 = P, P19 = P1, P29 = P2 as non
empty
Subset of (
TOP-REAL 2);
A5: (
[#] ((
TOP-REAL 2)
| P1))
= P1 by
PRE_TOPC:def 5;
consider h1, h2 such that
A6: h1 is
being_S-Seq and
A7: h2 is
being_S-Seq and
A8:
R^2-unit_square
= ((
L~ h1)
\/ (
L~ h2)) and
A9: ((
L~ h1)
/\ (
L~ h2))
=
{p00, p11} and
A10: (h1
/. 1)
= p00 and
A11: (h1
/. (
len h1))
= p11 and
A12: (h2
/. 1)
= p00 and
A13: (h2
/. (
len h2))
= p11 by
TOPREAL1: 24;
A14: (
len h2)
>= 2 by
A7,
TOPREAL1:def 8;
(
len h1)
>= 2 by
A6,
TOPREAL1:def 8;
then
reconsider Lh1 = (
L~ h1), Lh2 = (
L~ h2) as non
empty
Subset of (
TOP-REAL 2) by
A14,
TOPREAL1: 23;
set T1 = ((
TOP-REAL 2)
| Lh1), T2 = ((
TOP-REAL 2)
| Lh2), T = ((
TOP-REAL 2)
| RS);
A15: (
[#] T)
=
R^2-unit_square by
PRE_TOPC:def 5;
A16: (
[#] T2)
= (
L~ h2) by
PRE_TOPC:def 5;
then
A17: T2 is
SubSpace of T by
A8,
A15,
TOPMETR: 3,
XBOOLE_1: 7;
A18: (
[#] T1)
= (
L~ h1) by
PRE_TOPC:def 5;
then
A19: T1 is
SubSpace of T by
A8,
A15,
TOPMETR: 3,
XBOOLE_1: 7;
A20: (
[#] ((
TOP-REAL 2)
| P))
= P by
PRE_TOPC:def 5;
A21: (
[#] ((
TOP-REAL 2)
| P2))
= P2 by
PRE_TOPC:def 5;
then
A22: ((
TOP-REAL 2)
| P29) is
SubSpace of ((
TOP-REAL 2)
| P9) by
A3,
A20,
TOPMETR: 3,
XBOOLE_1: 7;
consider f2 be
Function of
I[01] , ((
TOP-REAL 2)
| P2) such that
A23: f2 is
being_homeomorphism and
A24: (f2
.
0 )
= p1 and
A25: (f2
. 1)
= p2 by
A2,
TOPREAL1:def 1;
A26: (
dom f2)
= the
carrier of
I[01] by
FUNCT_2:def 1;
P2
c= P by
A3,
XBOOLE_1: 7;
then (
rng f2)
c= the
carrier of ((
TOP-REAL 2)
| P) by
A21,
A20;
then
reconsider ff2 = f2 as
Function of
I[01] , ((
TOP-REAL 2)
| P9) by
A26,
RELSET_1: 4;
A27: (
dom ff2)
= the
carrier of
I[01] by
FUNCT_2:def 1;
then
A28:
0
in (
dom ff2) by
BORSUK_1: 40,
XXREAL_1: 1;
f2 is
continuous by
A23;
then
A29: ff2 is
continuous by
A22,
PRE_TOPC: 26;
A30: 1
in (
dom ff2) by
A27,
BORSUK_1: 40,
XXREAL_1: 1;
A31: (
[#] ((
TOP-REAL 2)
| P))
= P by
PRE_TOPC:def 5;
then
A32: ((
TOP-REAL 2)
| P19) is
SubSpace of ((
TOP-REAL 2)
| P9) by
A3,
A5,
TOPMETR: 3,
XBOOLE_1: 7;
consider f1 be
Function of
I[01] , ((
TOP-REAL 2)
| P1) such that
A33: f1 is
being_homeomorphism and
A34: (f1
.
0 )
= p1 and
A35: (f1
. 1)
= p2 by
A1,
TOPREAL1:def 1;
A36: (
dom f1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
P1
c= P by
A3,
XBOOLE_1: 7;
then (
rng f1)
c= the
carrier of ((
TOP-REAL 2)
| P) by
A5,
A31;
then
reconsider ff1 = f1 as
Function of
I[01] , ((
TOP-REAL 2)
| P9) by
A36,
RELSET_1: 4;
A37: (
dom f1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
A38:
I[01] is
compact by
HEINE: 4,
TOPMETR: 20;
f1 is
continuous by
A33;
then
A39: ff1 is
continuous by
A32,
PRE_TOPC: 26;
A40: f1 is
one-to-one by
A33;
reconsider L1 = (
L~ h1), L2 = (
L~ h2) as non
empty
Subset of (
TOP-REAL 2) by
A9;
L1
is_an_arc_of (p00,p11) by
A6,
A10,
A11,
TOPREAL1: 25;
then
consider g1 be
Function of
I[01] , ((
TOP-REAL 2)
| L1) such that
A41: g1 is
being_homeomorphism and
A42: (g1
.
0 )
= p00 and
A43: (g1
. 1)
= p11 by
TOPREAL1:def 1;
L2
is_an_arc_of (p00,p11) by
A7,
A12,
A13,
TOPREAL1: 25;
then
consider g2 be
Function of
I[01] , ((
TOP-REAL 2)
| L2) such that
A44: g2 is
being_homeomorphism and
A45: (g2
.
0 )
= p00 and
A46: (g2
. 1)
= p11 by
TOPREAL1:def 1;
R^2-unit_square
= (
[#] T) by
PRE_TOPC:def 5
.= the
carrier of T;
then
reconsider p00, p11 as
Point of T by
Lm28,
Lm29,
TOPREAL1: 14;
A47: T is
T_2 by
TOPMETR: 2;
set k1 = (ff1
* (g1
" )), k2 = (ff2
* (g2
" ));
reconsider g1 as
Function of
I[01] , ((
TOP-REAL 2)
| Lh1);
A48: g1 is
one-to-one by
A41;
A49: (
dom g1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
A50: (
rng g1)
= (
[#] T1) by
A41;
then g1 is
onto by
FUNCT_2:def 3;
then
A51: (g1
" )
= (g1 qua
Function
" ) by
A48,
TOPS_2:def 4;
then (
rng (g1
" ))
= (
dom g1) by
A48,
FUNCT_1: 33;
then
A52: (
rng k1)
= (
rng f1) by
A37,
A49,
RELAT_1: 28
.= P1 by
A33,
A5;
A53: (
dom g1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
then
A54:
0
in (
dom g1) by
BORSUK_1: 40,
XXREAL_1: 1;
then
A55:
0
= ((g1
" )
. p00) by
A42,
A48,
A51,
FUNCT_1: 32;
A56: (
dom (g1
" ))
= (
rng g1) by
A48,
A51,
FUNCT_1: 32;
then
A57: p00
in (
dom (g1
" )) by
A42,
A54,
FUNCT_1:def 3;
A58: 1
in (
dom g1) by
A53,
BORSUK_1: 40,
XXREAL_1: 1;
then
A59: p11
in (
dom (g1
" )) by
A43,
A56,
FUNCT_1:def 3;
reconsider g2 as
Function of
I[01] , ((
TOP-REAL 2)
| Lh2);
A60: g2 is
one-to-one by
A44;
A61: (
rng g2)
= (
[#] T2) by
A44;
then g2 is
onto by
FUNCT_2:def 3;
then
A62: (g2
" )
= (g2 qua
Function
" ) by
A60,
TOPS_2:def 4;
g2 is
continuous by
A44;
then
A63: T2 is
compact by
A38,
A61,
COMPTS_1: 14;
A64: (g2
" ) is
continuous by
A44;
g1 is
continuous by
A41;
then
A65: T1 is
compact by
A38,
A50,
COMPTS_1: 14;
A66: f2 is
one-to-one by
A23;
A67: (
dom g2)
= the
carrier of
I[01] by
FUNCT_2:def 1;
then
A68:
0
in (
dom g2) by
BORSUK_1: 40,
XXREAL_1: 1;
then
A69: p00
in (
rng g2) by
A45,
FUNCT_1:def 3;
then
A70: p00
in (
dom (g2
" )) by
A60,
A62,
FUNCT_1: 32;
((g2
" )
. p00)
in (
dom ff2) by
A45,
A60,
A62,
A53,
A67,
A27,
A54,
FUNCT_1: 32;
then
A71: p00
in (
dom (ff2
* (g2
" ))) by
A70,
FUNCT_1: 11;
A72: (
dom ff1)
= the
carrier of
I[01] by
FUNCT_2:def 1;
then ((g1
" )
. p00)
in (
dom ff1) by
A42,
A48,
A51,
A53,
A54,
FUNCT_1: 32;
then p00
in (
dom (ff1
* (g1
" ))) by
A57,
FUNCT_1: 11;
then
A73: (k1
. p00)
= (ff1
. ((g1
" )
. p00)) by
FUNCT_1: 12
.= p1 by
A34,
A42,
A48,
A51,
A54,
FUNCT_1: 32;
then
A74: (k1
. p00)
= (ff2
. ((g2
" )
. p00)) by
A24,
A45,
A60,
A62,
A68,
FUNCT_1: 32
.= (k2
. p00) by
A71,
FUNCT_1: 12;
A75: 1
in (
dom g2) by
A67,
BORSUK_1: 40,
XXREAL_1: 1;
then
A76: 1
= ((g2
" )
. p11) by
A46,
A60,
A62,
FUNCT_1: 32;
A77: (
dom (g2
" ))
= (
rng g2) by
A60,
A62,
FUNCT_1: 32;
then
A78: p11
in (
dom (g2
" )) by
A46,
A75,
FUNCT_1:def 3;
((g2
" )
. p11)
in (
dom ff2) by
A46,
A60,
A62,
A53,
A67,
A27,
A58,
FUNCT_1: 32;
then
A79: p11
in (
dom (ff2
* (g2
" ))) by
A78,
FUNCT_1: 11;
((g1
" )
. p11)
in (
dom ff1) by
A43,
A48,
A51,
A53,
A72,
A58,
FUNCT_1: 32;
then p11
in (
dom (ff1
* (g1
" ))) by
A59,
FUNCT_1: 11;
then
A80: (k1
. p11)
= (ff1
. ((g1
" )
. p11)) by
FUNCT_1: 12
.= p2 by
A35,
A43,
A48,
A51,
A58,
FUNCT_1: 32;
then
A81: (k1
. p11)
= (ff2
. ((g2
" )
. p11)) by
A25,
A46,
A60,
A62,
A75,
FUNCT_1: 32
.= (k2
. p11) by
A79,
FUNCT_1: 12;
(g1
" ) is
continuous by
A41;
then
reconsider h = (k1
+* k2) as
continuous
Function of T, ((
TOP-REAL 2)
| P) by
A8,
A9,
A39,
A29,
A18,
A16,
A15,
A65,
A63,
A47,
A64,
A74,
A81,
A19,
A17,
COMPTS_1: 21;
A82: 1
= ((g1
" )
. p11) by
A43,
A48,
A51,
A58,
FUNCT_1: 32;
A83: (
rng (g2
" ))
= (
dom g2) by
A60,
A62,
FUNCT_1: 33;
then
A84: (
rng k2)
= (
rng f2) by
A67,
A27,
RELAT_1: 28
.= (
[#] ((
TOP-REAL 2)
| P2)) by
A23
.= P2 by
PRE_TOPC:def 5;
A85:
0
= ((g2
" )
. p00) by
A45,
A60,
A62,
A68,
FUNCT_1: 32;
now
let x1,x2 be
set;
assume that
A86: x1
in (
dom k2) and
A87: x2
in ((
dom k1)
\ (
dom k2));
A88: x1
in (
dom (g2
" )) by
A86,
FUNCT_1: 11;
A89: (k2
. x1)
in P2 by
A84,
A86,
FUNCT_1:def 3;
A90: x2
in (
dom k1) by
A87,
XBOOLE_0:def 5;
then
A91: x2
in (
dom (g1
" )) by
FUNCT_1: 11;
assume
A92: (k2
. x1)
= (k1
. x2);
then (k2
. x1)
in P1 by
A52,
A90,
FUNCT_1:def 3;
then
A93: (k2
. x1)
in (P1
/\ P2) by
A89,
XBOOLE_0:def 4;
per cases by
A4,
A93,
TARSKI:def 2;
suppose
A94: (k2
. x1)
= p1;
A95: ((g1
" )
. x2)
in (
dom ff1) by
A90,
FUNCT_1: 11;
p1
= (ff1
. ((g1
" )
. x2)) by
A92,
A90,
A94,
FUNCT_1: 12;
then
A96: ((g1
" )
. x2)
=
0 by
A34,
A72,
A28,
A40,
A95,
FUNCT_1:def 4;
A97: p00
in (
dom (g2
" )) by
A60,
A62,
A69,
FUNCT_1: 32;
A98: ((g2
" )
. x1)
in (
dom ff2) by
A86,
FUNCT_1: 11;
p1
= (ff2
. ((g2
" )
. x1)) by
A86,
A94,
FUNCT_1: 12;
then ((g2
" )
. x1)
=
0 by
A24,
A28,
A66,
A98,
FUNCT_1:def 4;
then
A99: x1
= p00 by
A60,
A62,
A85,
A88,
A97,
FUNCT_1:def 4;
p00
in (
dom (g1
" )) by
A42,
A53,
A28,
A56,
FUNCT_1:def 3;
then x2
in (
dom k2) by
A48,
A51,
A55,
A86,
A91,
A99,
A96,
FUNCT_1:def 4;
hence contradiction by
A87,
XBOOLE_0:def 5;
end;
suppose
A100: (k2
. x1)
= p2;
A101: ((g1
" )
. x2)
in (
dom ff1) by
A90,
FUNCT_1: 11;
p2
= (ff1
. ((g1
" )
. x2)) by
A92,
A90,
A100,
FUNCT_1: 12;
then
A102: ((g1
" )
. x2)
= 1 by
A35,
A72,
A30,
A40,
A101,
FUNCT_1:def 4;
A103: p11
in (
dom (g2
" )) by
A46,
A67,
A77,
A30,
FUNCT_1:def 3;
A104: ((g2
" )
. x1)
in (
dom ff2) by
A86,
FUNCT_1: 11;
p2
= (ff2
. ((g2
" )
. x1)) by
A86,
A100,
FUNCT_1: 12;
then ((g2
" )
. x1)
= 1 by
A25,
A30,
A66,
A104,
FUNCT_1:def 4;
then
A105: x1
= p11 by
A60,
A62,
A76,
A88,
A103,
FUNCT_1:def 4;
p11
in (
dom (g1
" )) by
A43,
A53,
A56,
A30,
FUNCT_1:def 3;
then x2
in (
dom k2) by
A48,
A51,
A82,
A86,
A91,
A105,
A102,
FUNCT_1:def 4;
hence contradiction by
A87,
XBOOLE_0:def 5;
end;
end;
then
A106: h is
one-to-one by
A48,
A60,
A62,
A51,
A40,
A66,
TOPMETR2: 1;
A107: ((
TOP-REAL 2)
| P9) is
T_2 by
TOPMETR: 2;
A108: (
dom k2)
= (
dom (g2
" )) by
A27,
A83,
RELAT_1: 27;
(k1
.: ((
dom k1)
/\ (
dom k2)))
c= (
rng k2)
proof
let a be
object;
A109: (
dom k2)
= the
carrier of T2 by
FUNCT_2:def 1;
assume a
in (k1
.: ((
dom k1)
/\ (
dom k2)));
then
A110: ex x be
object st x
in (
dom k1) & x
in ((
dom k1)
/\ (
dom k2)) & a
= (k1
. x) by
FUNCT_1:def 6;
(
dom k1)
= the
carrier of T1 by
FUNCT_2:def 1;
then a
= p1 or a
= p2 by
A9,
A18,
A16,
A73,
A80,
A110,
A109,
TARSKI:def 2;
hence thesis by
A70,
A73,
A74,
A78,
A80,
A81,
A108,
FUNCT_1:def 3;
end;
then
A111: (
rng h)
= (
[#] ((
TOP-REAL 2)
| P9)) by
A3,
A31,
A52,
A84,
TOPMETR2: 2;
reconsider h as
Function of ((
TOP-REAL 2)
|
R^2-unit_square ), ((
TOP-REAL 2)
| P);
take h;
T is
compact by
Th2,
COMPTS_1: 3;
hence thesis by
A107,
A111,
A106,
COMPTS_1: 17;
end;
theorem ::
TOPREAL2:5
Th5: for P be non
empty
Subset of (
TOP-REAL 2) holds P is
being_simple_closed_curve iff (ex p1, p2 st p1
<> p2 & p1
in P & p2
in P) & for p1, p2 st p1
<> p2 & p1
in P & p2
in P holds ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
let P be non
empty
Subset of (
TOP-REAL 2);
thus P is
being_simple_closed_curve implies (ex p1, p2 st p1
<> p2 & p1
in P & p2
in P) & for p1, p2 st p1
<> p2 & p1
in P & p2
in P holds ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
assume
A1: P is
being_simple_closed_curve;
then
consider f be
Function of ((
TOP-REAL 2)
|
R^2-unit_square ), ((
TOP-REAL 2)
| P) such that
A2: f is
being_homeomorphism;
A3: (
dom f)
= (
[#] ((
TOP-REAL 2)
|
R^2-unit_square )) by
A2;
A4: (
[#] ((
TOP-REAL 2)
| P))
c= (
[#] (
TOP-REAL 2)) by
PRE_TOPC:def 4;
A5: f is
continuous by
A2;
thus ex p1, p2 st p1
<> p2 & p1
in P & p2
in P by
A1,
Th4;
set RS =
R^2-unit_square ;
let p1, p2;
assume that
A6: p1
<> p2 and
A7: p1
in P and
A8: p2
in P;
A9: (
[#] ((
TOP-REAL 2)
|
R^2-unit_square ))
=
R^2-unit_square by
PRE_TOPC:def 5;
set q1 = ((f
" )
. p1), q2 = ((f
" )
. p2);
A10: (
[#] ((
TOP-REAL 2)
| RS))
c= (
[#] (
TOP-REAL 2)) by
PRE_TOPC:def 4;
A11:
I[01] is
compact by
HEINE: 4,
TOPMETR: 20;
A12: f is
one-to-one by
A2;
A13: (
rng f)
= (
[#] ((
TOP-REAL 2)
| P)) by
A2;
then f is
onto by
FUNCT_2:def 3;
then
A14: (f
" )
= (f qua
Function
" ) by
A12,
TOPS_2:def 4;
then
A15: (
rng (f
" ))
= (
dom f) by
A12,
FUNCT_1: 33;
A16: (
dom (f
" ))
= (
rng f) by
A12,
A14,
FUNCT_1: 32;
then
A17: p1
in (
dom (f
" )) by
A7,
A13,
PRE_TOPC:def 5;
A18: p2
in (
dom (f
" )) by
A8,
A13,
A16,
PRE_TOPC:def 5;
reconsider f as
Function of ((
TOP-REAL 2)
| RS), ((
TOP-REAL 2)
| P);
A19: q1
in (
rng (f
" )) by
A17,
FUNCT_1:def 3;
A20: q2
in (
rng (f
" )) by
A18,
FUNCT_1:def 3;
reconsider q1, q2 as
Point of (
TOP-REAL 2) by
A10,
A19,
A20;
A21: q1
<> q2 by
A6,
A12,
A14,
A17,
A18,
FUNCT_1:def 4;
A22: (
dom f)
= the
carrier of ((
TOP-REAL 2)
|
R^2-unit_square ) by
FUNCT_2:def 1;
then
A23: q2
in
R^2-unit_square by
A15,
A18,
A9,
FUNCT_1:def 3;
A24: p1
= (f
. q1) by
A12,
A14,
A16,
A17,
FUNCT_1: 35;
q1
in
R^2-unit_square by
A15,
A17,
A22,
A9,
FUNCT_1:def 3;
then
consider Q1,Q2 be non
empty
Subset of (
TOP-REAL 2) such that
A25: Q1
is_an_arc_of (q1,q2) and
A26: Q2
is_an_arc_of (q1,q2) and
A27:
R^2-unit_square
= (Q1
\/ Q2) and
A28: (Q1
/\ Q2)
=
{q1, q2} by
A21,
A23,
Th1;
A29: Q2
c= (
dom f) by
A22,
A9,
A27,
XBOOLE_1: 7;
set P1 = (f
.: Q1), P2 = (f
.: Q2);
Q1
c= (
dom f) by
A22,
A9,
A27,
XBOOLE_1: 7;
then
reconsider P1, P2 as non
empty
Subset of (
TOP-REAL 2) by
A29,
A4,
XBOOLE_1: 1;
A30: (
rng (f
| Q1))
= P1 by
RELAT_1: 115
.= (
[#] ((
TOP-REAL 2)
| P1)) by
PRE_TOPC:def 5
.= the
carrier of ((
TOP-REAL 2)
| P1);
(
dom (f
| Q1))
= (
R^2-unit_square
/\ Q1) by
A22,
A9,
RELAT_1: 61
.= Q1 by
A27,
XBOOLE_1: 21
.= (
[#] ((
TOP-REAL 2)
| Q1)) by
PRE_TOPC:def 5;
then
reconsider F1 = (f
| Q1) as
Function of ((
TOP-REAL 2)
| Q1), ((
TOP-REAL 2)
| P1) by
A30,
FUNCT_2:def 1,
RELSET_1: 4;
A31: (f
" P1)
c= Q1 by
A12,
FUNCT_1: 82;
(
[#] ((
TOP-REAL 2)
| Q1))
= Q1 by
PRE_TOPC:def 5;
then
A32: ((
TOP-REAL 2)
| Q1) is
SubSpace of ((
TOP-REAL 2)
|
R^2-unit_square ) by
A9,
A27,
TOPMETR: 3,
XBOOLE_1: 7;
Q1
c= (f
" P1) by
A22,
A9,
A27,
FUNCT_1: 76,
XBOOLE_1: 7;
then
A33: (f
" P1)
= Q1 by
A31,
XBOOLE_0:def 10;
for R be
Subset of ((
TOP-REAL 2)
| P1) st R is
closed holds (F1
" R) is
closed
proof
let R be
Subset of ((
TOP-REAL 2)
| P1);
assume R is
closed;
then
consider S1 be
Subset of (
TOP-REAL 2) such that
A34: S1 is
closed and
A35: R
= (S1
/\ (
[#] ((
TOP-REAL 2)
| P1))) by
PRE_TOPC: 13;
(S1
/\ (
rng f)) is
Subset of ((
TOP-REAL 2)
| P);
then
reconsider S2 = ((
rng f)
/\ S1) as
Subset of ((
TOP-REAL 2)
| P);
S2 is
closed by
A13,
A34,
PRE_TOPC: 13;
then
A36: (f
" S2) is
closed by
A5;
(F1
" R)
= (Q1
/\ (f
" R)) by
FUNCT_1: 70
.= (Q1
/\ ((f
" S1)
/\ (f
" (
[#] ((
TOP-REAL 2)
| P1))))) by
A35,
FUNCT_1: 68
.= (((f
" S1)
/\ Q1)
/\ Q1) by
A33,
PRE_TOPC:def 5
.= ((f
" S1)
/\ (Q1
/\ Q1)) by
XBOOLE_1: 16
.= ((f
" S1)
/\ (
[#] ((
TOP-REAL 2)
| Q1))) by
PRE_TOPC:def 5
.= ((f
" ((
rng f)
/\ S1))
/\ (
[#] ((
TOP-REAL 2)
| Q1))) by
RELAT_1: 133;
hence thesis by
A32,
A36,
PRE_TOPC: 13;
end;
then
A37: F1 is
continuous;
reconsider Q19 = Q1, Q29 = Q2 as non
empty
Subset of (
TOP-REAL 2);
consider ff be
Function of
I[01] , ((
TOP-REAL 2)
| Q1) such that
A38: ff is
being_homeomorphism and (ff
.
0 )
= q1 and (ff
. 1)
= q2 by
A25,
TOPREAL1:def 1;
A39: (
rng ff)
= (
[#] ((
TOP-REAL 2)
| Q1)) by
A38;
A40: (
rng (f
| Q2))
= P2 by
RELAT_1: 115
.= (
[#] ((
TOP-REAL 2)
| P2)) by
PRE_TOPC:def 5
.= the
carrier of ((
TOP-REAL 2)
| P2);
A41: p2
= (f
. q2) by
A12,
A14,
A16,
A18,
FUNCT_1: 35;
(
dom (f
| Q2))
= (
R^2-unit_square
/\ Q2) by
A22,
A9,
RELAT_1: 61
.= Q2 by
A27,
XBOOLE_1: 21
.= (
[#] ((
TOP-REAL 2)
| Q2)) by
PRE_TOPC:def 5;
then
reconsider F2 = (f
| Q2) as
Function of ((
TOP-REAL 2)
| Q2), ((
TOP-REAL 2)
| P2) by
A40,
FUNCT_2:def 1,
RELSET_1: 4;
A42: (f
" P2)
c= Q2 by
A12,
FUNCT_1: 82;
(
[#] ((
TOP-REAL 2)
| Q2))
= Q2 by
PRE_TOPC:def 5;
then
A43: ((
TOP-REAL 2)
| Q2) is
SubSpace of ((
TOP-REAL 2)
|
R^2-unit_square ) by
A9,
A27,
TOPMETR: 3,
XBOOLE_1: 7;
Q2
c= (f
" P2) by
A22,
A9,
A27,
FUNCT_1: 76,
XBOOLE_1: 7;
then
A44: (f
" P2)
= Q2 by
A42,
XBOOLE_0:def 10;
for R be
Subset of ((
TOP-REAL 2)
| P2) st R is
closed holds (F2
" R) is
closed
proof
let R be
Subset of ((
TOP-REAL 2)
| P2);
assume R is
closed;
then
consider S1 be
Subset of (
TOP-REAL 2) such that
A45: S1 is
closed and
A46: R
= (S1
/\ (
[#] ((
TOP-REAL 2)
| P2))) by
PRE_TOPC: 13;
(S1
/\ (
rng f)) is
Subset of ((
TOP-REAL 2)
| P);
then
reconsider S2 = ((
rng f)
/\ S1) as
Subset of ((
TOP-REAL 2)
| P);
S2 is
closed by
A13,
A45,
PRE_TOPC: 13;
then
A47: (f
" S2) is
closed by
A5;
(F2
" R)
= (Q2
/\ (f
" R)) by
FUNCT_1: 70
.= (Q2
/\ ((f
" S1)
/\ (f
" (
[#] ((
TOP-REAL 2)
| P2))))) by
A46,
FUNCT_1: 68
.= (((f
" S1)
/\ Q2)
/\ Q2) by
A44,
PRE_TOPC:def 5
.= ((f
" S1)
/\ (Q2
/\ Q2)) by
XBOOLE_1: 16
.= ((f
" S1)
/\ (
[#] ((
TOP-REAL 2)
| Q2))) by
PRE_TOPC:def 5
.= ((f
" ((
rng f)
/\ S1))
/\ (
[#] ((
TOP-REAL 2)
| Q2))) by
RELAT_1: 133;
hence thesis by
A43,
A47,
PRE_TOPC: 13;
end;
then
A48: F2 is
continuous;
A49: q2
in
{q1, q2} by
TARSKI:def 2;
A50: q1
in
{q1, q2} by
TARSKI:def 2;
A51: q1
in
{q1, q2} by
TARSKI:def 2;
{q1, q2}
c= Q1 by
A28,
XBOOLE_1: 17;
then
A52: q1
in ((
dom f)
/\ Q1) by
A15,
A19,
A51,
XBOOLE_0:def 4;
take P1, P2;
A53: ((
TOP-REAL 2)
| P1) is
T_2 by
TOPMETR: 2;
A54: q2
in
{q1, q2} by
TARSKI:def 2;
{q1, q2}
c= Q1 by
A28,
XBOOLE_1: 17;
then
A55: q2
in ((
dom f)
/\ Q1) by
A15,
A20,
A54,
XBOOLE_0:def 4;
A56: p2
= (f
. q2) by
A12,
A14,
A16,
A18,
FUNCT_1: 35
.= (F1
. q2) by
A55,
FUNCT_1: 48;
A57: (
rng F1)
= (
[#] ((
TOP-REAL 2)
| P1)) by
A30;
ff is
continuous by
A38;
then
A58: ((
TOP-REAL 2)
| Q19) is
compact by
A11,
A39,
COMPTS_1: 14;
A59: F1 is
one-to-one by
A12,
FUNCT_1: 52;
p1
= (f
. q1) by
A12,
A14,
A16,
A17,
FUNCT_1: 35
.= (F1
. q1) by
A52,
FUNCT_1: 48;
hence P1
is_an_arc_of (p1,p2) by
A25,
A57,
A59,
A37,
A58,
A53,
A56,
Th3,
COMPTS_1: 17;
A60: ((
TOP-REAL 2)
| P2) is
T_2 by
TOPMETR: 2;
consider ff be
Function of
I[01] , ((
TOP-REAL 2)
| Q2) such that
A61: ff is
being_homeomorphism and (ff
.
0 )
= q1 and (ff
. 1)
= q2 by
A26,
TOPREAL1:def 1;
A62: (
rng ff)
= (
[#] ((
TOP-REAL 2)
| Q2)) by
A61;
{q1, q2}
c= Q2 by
A28,
XBOOLE_1: 17;
then q1
in ((
dom f)
/\ Q2) by
A15,
A19,
A50,
XBOOLE_0:def 4;
then
A63: p1
= (F2
. q1) by
A24,
FUNCT_1: 48;
A64: F2 is
one-to-one by
A12,
FUNCT_1: 52;
{q1, q2}
c= Q2 by
A28,
XBOOLE_1: 17;
then q2
in ((
dom f)
/\ Q2) by
A15,
A20,
A49,
XBOOLE_0:def 4;
then
A65: p2
= (F2
. q2) by
A41,
FUNCT_1: 48;
ff is
continuous by
A61;
then
A66: ((
TOP-REAL 2)
| Q29) is
compact by
A11,
A62,
COMPTS_1: 14;
(
rng F2)
= (
[#] ((
TOP-REAL 2)
| P2)) by
A40;
hence P2
is_an_arc_of (p1,p2) by
A26,
A64,
A48,
A66,
A60,
A63,
A65,
Th3,
COMPTS_1: 17;
(
[#] ((
TOP-REAL 2)
| P))
= P by
PRE_TOPC:def 5;
hence P
= (f
.: (Q1
\/ Q2)) by
A13,
A3,
A9,
A27,
RELAT_1: 113
.= (P1
\/ P2) by
RELAT_1: 120;
thus (P1
/\ P2)
= (f
.: (Q1
/\ Q2)) by
A12,
FUNCT_1: 62
.= (f
.: (
{q1}
\/
{q2})) by
A28,
ENUMSET1: 1
.= ((
Im (f,q1))
\/ (
Im (f,q2))) by
RELAT_1: 120
.= (
{p1}
\/ (
Im (f,q2))) by
A15,
A19,
A24,
FUNCT_1: 59
.= (
{p1}
\/
{p2}) by
A15,
A20,
A41,
FUNCT_1: 59
.=
{p1, p2} by
ENUMSET1: 1;
end;
given p1, p2 such that
A67: p1
<> p2 and
A68: p1
in P and
A69: p2
in P;
assume for p1, p2 st p1
<> p2 & p1
in P & p2
in P holds ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2};
then ex P1,P2 be non
empty
Subset of (
TOP-REAL 2) st P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2} by
A67,
A68,
A69;
hence thesis by
Lm34;
end;
theorem ::
TOPREAL2:6
for P be non
empty
Subset of (
TOP-REAL 2) holds P is
being_simple_closed_curve iff ex p1,p2 be
Point of (
TOP-REAL 2), P1,P2 be non
empty
Subset of (
TOP-REAL 2) st p1
<> p2 & p1
in P & p2
in P & P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2}
proof
let P be non
empty
Subset of (
TOP-REAL 2);
hereby
assume
A1: P is
being_simple_closed_curve;
then
consider p1, p2 such that
A2: p1
<> p2 and
A3: p1
in P and
A4: p2
in P by
Th5;
consider P1,P2 be non
empty
Subset of (
TOP-REAL 2) such that
A5: P1
is_an_arc_of (p1,p2) and
A6: P2
is_an_arc_of (p1,p2) and
A7: P
= (P1
\/ P2) and
A8: (P1
/\ P2)
=
{p1, p2} by
A1,
A2,
A3,
A4,
Th5;
take p1, p2, P1, P2;
thus p1
<> p2 & p1
in P & p2
in P & P1
is_an_arc_of (p1,p2) & P2
is_an_arc_of (p1,p2) & P
= (P1
\/ P2) & (P1
/\ P2)
=
{p1, p2} by
A2,
A3,
A4,
A5,
A6,
A7,
A8;
end;
thus thesis by
Lm34;
end;
Lm35: for S be
1-sorted, T be
1-sorted, f be
Function of S, T st S is
empty & (
rng f)
= (
[#] T) holds T is
empty
proof
let S be
1-sorted, T be
1-sorted, f be
Function of S, T such that
A1: S is
empty and
A2: (
rng f)
= (
[#] T);
assume T is non
empty;
then
reconsider T as non
empty
1-sorted;
consider y be
object such that
A3: y
in the
carrier of T by
XBOOLE_0:def 1;
ex x be
object st x
in (
dom f) & (f
. x)
= y by
A2,
A3,
FUNCT_1:def 3;
hence contradiction by
A1;
end;
Lm36: for S be
1-sorted, T be
1-sorted, f be
Function of S, T st T is
empty & (
dom f)
= (
[#] S) holds S is
empty
proof
let S be
1-sorted, T be
1-sorted, f be
Function of S, T such that
A1: T is
empty and
A2: (
dom f)
= (
[#] S);
assume S is non
empty;
then
reconsider S as non
empty
1-sorted;
consider x be
object such that
A3: x
in the
carrier of S by
XBOOLE_0:def 1;
(f
. x)
in (
rng f) by
A2,
A3,
FUNCT_1:def 3;
hence thesis by
A1;
end;
Lm37: for S,T be
TopStruct st ex f be
Function of S, T st f is
being_homeomorphism holds S is
empty iff T is
empty by
Lm35,
Lm36;
registration
cluster
being_simple_closed_curve -> non
empty
compact for
Subset of (
TOP-REAL 2);
coherence
proof
let P be
Subset of (
TOP-REAL 2);
given f be
Function of ((
TOP-REAL 2)
|
R^2-unit_square ), ((
TOP-REAL 2)
| P) such that
A1: f is
being_homeomorphism;
thus P is non
empty by
A1,
Lm37;
A2: (
rng f)
= (
[#] ((
TOP-REAL 2)
| P)) by
A1;
reconsider R = P as non
empty
Subset of (
TOP-REAL 2) by
A1,
Lm37;
A3: f is
continuous by
A1;
((
TOP-REAL 2)
|
R^2-unit_square ) is
compact by
Th2,
COMPTS_1: 3;
then ((
TOP-REAL 2)
| R) is
compact by
A3,
A2,
COMPTS_1: 14;
hence thesis by
COMPTS_1: 3;
end;
end