let p1, p2 be Point of (TOP-REAL 2); for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `2 <= 0 & p1 <> W-min P holds
p1 `1 > p2 `1
let P be non empty compact Subset of (TOP-REAL 2); ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `2 <= 0 & p1 <> W-min P implies p1 `1 > p2 `1 )
assume that
A1:
P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 }
and
A2:
LE p1,p2,P
and
A3:
p1 <> p2
and
A4:
p1 `2 <= 0
and
A5:
p1 <> W-min P
; p1 `1 > p2 `1
A6:
P is being_simple_closed_curve
by A1, JGRAPH_3:26;
then A7:
p2 in P
by A2, JORDAN7:5;
set P4 = Lower_Arc P;
A8:
Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) }
by A1, Th35;
A9:
(Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)}
by A6, JORDAN6:def 9;
A10:
p1 in P
by A2, A6, JORDAN7:5;
now ( p1 in Upper_Arc P implies p1 `1 > p2 `1 )assume A11:
p1 in Upper_Arc P
;
p1 `1 > p2 `1
p1 in Lower_Arc P
by A4, A10, A8;
then
p1 in {(W-min P),(E-max P)}
by A9, A11, XBOOLE_0:def 4;
then
(
p1 = W-min P or
p1 = E-max P )
by TARSKI:def 2;
then A12:
p1 = |[1,0]|
by A1, A5, Th30;
then A13:
p1 `1 = 1
by EUCLID:52;
A14:
ex
p9 being
Point of
(TOP-REAL 2) st
(
p9 = p2 &
|.p9.| = 1 )
by A1, A7;
p2 `1 <= 1
by A14, Th1;
hence
p1 `1 > p2 `1
by A13, A15, XXREAL_0:1;
verum end;
then A17:
( ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P ) or p1 `1 > p2 `1 )
by A2;
consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) such that
A18:
f is being_homeomorphism
and
A19:
for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 )
and
A20:
( f . 0 = E-max P & f . 1 = W-min P )
by A1, Th42;
A21: rng f =
[#] ((TOP-REAL 2) | (Lower_Arc P))
by A18, TOPS_2:def 5
.=
Lower_Arc P
by PRE_TOPC:def 5
;
now ( ( not p1 `1 > p2 `1 & p1 `1 > p2 `1 ) or ( p1 `1 > p2 `1 & p1 `1 > p2 `1 ) )per cases
( not p1 `1 > p2 `1 or p1 `1 > p2 `1 )
;
case A22:
not
p1 `1 > p2 `1
;
p1 `1 > p2 `1 then consider x1 being
object such that A23:
x1 in dom f
and A24:
p1 = f . x1
by A17, A21, FUNCT_1:def 3;
consider x2 being
object such that A25:
x2 in dom f
and A26:
p2 = f . x2
by A17, A21, A22, FUNCT_1:def 3;
A27:
dom f =
[#] I[01]
by A18, TOPS_2:def 5
.=
[.0,1.]
by BORSUK_1:40
;
reconsider r22 =
x2 as
Real by A25;
A28:
(
0 <= r22 &
r22 <= 1 )
by A25, A27, XXREAL_1:1;
reconsider r11 =
x1 as
Real by A23;
A29:
(
r11 < r22 iff
p1 `1 > p2 `1 )
by A19, A23, A24, A25, A26, A27;
r11 <= 1
by A23, A27, XXREAL_1:1;
then
(
r11 <= r22 or
p1 `1 > p2 `1 )
by A17, A18, A20, A24, A26, A28, JORDAN5C:def 3;
hence
p1 `1 > p2 `1
by A3, A24, A26, A29, XXREAL_0:1;
verum end;
hence
p1 `1 > p2 `1
; verum