let C be being_simple_closed_curve Subset of (TOP-REAL 2); Y-InitStart C > 1
set m = ApproxIndex C;
A1:
(X-SpanStart (C,(ApproxIndex C))) -' 1 <= len (Gauge (C,(ApproxIndex C)))
by JORDAN1H:50;
assume A2:
Y-InitStart C <= 1
; contradiction
per cases
( Y-InitStart C = 0 or Y-InitStart C = 1 )
by A2, NAT_1:25;
suppose A3:
Y-InitStart C = 0
;
contradictionA4:
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
0)
c= UBD C
by A1, JORDAN1A:49;
0 <= width (Gauge (C,(ApproxIndex C)))
;
then A5:
not
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
0) is
empty
by A1, JORDAN1A:24;
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
0)
c= BDD C
by A3, Def2;
hence
contradiction
by A5, A4, JORDAN2C:24, XBOOLE_1:68;
verum end; suppose
Y-InitStart C = 1
;
contradictionthen A6:
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),1)
c= BDD C
by Def2;
set i1 =
X-SpanStart (
C,
(ApproxIndex C));
A7:
(X-SpanStart (C,(ApproxIndex C))) -' 1
<= X-SpanStart (
C,
(ApproxIndex C))
by NAT_D:44;
X-SpanStart (
C,
(ApproxIndex C))
< len (Gauge (C,(ApproxIndex C)))
by JORDAN1H:49;
then A8:
(X-SpanStart (C,(ApproxIndex C))) -' 1
< len (Gauge (C,(ApproxIndex C)))
by A7, XXREAL_0:2;
BDD C c= C `
by JORDAN2C:25;
then A9:
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),1)
c= C `
by A6;
UBD C is_outside_component_of C
by JORDAN2C:68;
then A10:
UBD C is_a_component_of C `
by JORDAN2C:def 3;
A11:
width (Gauge (C,(ApproxIndex C))) <> 0
by MATRIX_0:def 10;
then A12:
0 + 1
<= width (Gauge (C,(ApproxIndex C)))
by NAT_1:14;
then A13:
not
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),1) is
empty
by A1, JORDAN1A:24;
1
<= (X-SpanStart (C,(ApproxIndex C))) -' 1
by JORDAN1H:50;
then
(cell ((Gauge (C,(ApproxIndex C))),((X-SpanStart (C,(ApproxIndex C))) -' 1),0)) /\ (cell ((Gauge (C,(ApproxIndex C))),((X-SpanStart (C,(ApproxIndex C))) -' 1),(0 + 1))) = LSeg (
((Gauge (C,(ApproxIndex C))) * (((X-SpanStart (C,(ApproxIndex C))) -' 1),(0 + 1))),
((Gauge (C,(ApproxIndex C))) * ((((X-SpanStart (C,(ApproxIndex C))) -' 1) + 1),(0 + 1))))
by A11, A8, GOBOARD5:26;
then A14:
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
0)
meets cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
(0 + 1))
;
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),
0)
c= UBD C
by A1, JORDAN1A:49;
then
cell (
(Gauge (C,(ApproxIndex C))),
((X-SpanStart (C,(ApproxIndex C))) -' 1),1)
c= UBD C
by A12, A8, A14, A10, A9, GOBOARD9:4, JORDAN1A:25;
hence
contradiction
by A6, A13, JORDAN2C:24, XBOOLE_1:68;
verum end;