let C be being_simple_closed_curve Subset of (TOP-REAL 2); :: thesis: (Y-InitStart C) + 1 < width (Gauge C,(ApproxIndex C))
set m = ApproxIndex C;
A1: (X-SpanStart C,(ApproxIndex C)) -' 1 <= len (Gauge C,(ApproxIndex C)) by JORDAN1H:59;
assume (Y-InitStart C) + 1 >= width (Gauge C,(ApproxIndex C)) ; :: thesis: contradiction
then A2: ( (Y-InitStart C) + 1 > width (Gauge C,(ApproxIndex C)) or (Y-InitStart C) + 1 = width (Gauge C,(ApproxIndex C)) ) by XXREAL_0:1;
A3: ( Y-InitStart C < width (Gauge C,(ApproxIndex C)) or Y-InitStart C = width (Gauge C,(ApproxIndex C)) ) by Def2;

per cases ( Y-InitStart C = width (Gauge C,(ApproxIndex C)) or (Y-InitStart C) + 1 = width (Gauge C,(ApproxIndex C)) ) by A2, A3, NAT_1:13;

suppose Y-InitStart C = width (Gauge C,(ApproxIndex C)) ; :: thesis: contradiction

hence contradiction by Def2; :: thesis: verum

end;

suppose (Y-InitStart C) + 1 = width (Gauge C,(ApproxIndex C)) ; :: thesis: contradiction

then Y-InitStart C = (width (Gauge C,(ApproxIndex C))) -' 1 by NAT_D:34;
then A4: cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1) c= BDD C by Def2;
BDD C c= C ` by JORDAN2C:29;
then A5: cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1) c= C ` by A4, XBOOLE_1:1;
A6: cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),(width (Gauge C,(ApproxIndex C))) c= UBD C by A1, JORDAN1A:71;
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:58;
then A8: (X-SpanStart C,(ApproxIndex C)) -' 1 < len (Gauge C,(ApproxIndex C)) by A7, XXREAL_0:2;
UBD C is_outside_component_of C by JORDAN2C:76;
then A9: UBD C is_a_component_of C ` by JORDAN2C:def 4;
(width (Gauge C,(ApproxIndex C))) -' 1 <= width (Gauge C,(ApproxIndex C)) by NAT_D:44;
then A10: not cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1) is empty by A1, JORDAN1A:45;
A11: (width (Gauge C,(ApproxIndex C))) - 1 < width (Gauge C,(ApproxIndex C)) by XREAL_1:148;
A12: 1 <= (X-SpanStart C,(ApproxIndex C)) -' 1 by JORDAN1H:59;
A13: width (Gauge C,(ApproxIndex C)) <> 0 by GOBOARD1:def 5;
then ((width (Gauge C,(ApproxIndex C))) -' 1) + 1 = width (Gauge C,(ApproxIndex C)) by NAT_1:14, XREAL_1:237;
then (cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),(width (Gauge C,(ApproxIndex C)))) /\ (cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1)) = LSeg ((Gauge C,(ApproxIndex C)) * ((X-SpanStart C,(ApproxIndex C)) -' 1),(width (Gauge C,(ApproxIndex C)))),((Gauge C,(ApproxIndex C)) * (((X-SpanStart C,(ApproxIndex C)) -' 1) + 1),(width (Gauge C,(ApproxIndex C)))) by A8, A11, A12, GOBOARD5:27;
then A14: cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),(width (Gauge C,(ApproxIndex C))) meets cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1) by XBOOLE_0:def 7;
(width (Gauge C,(ApproxIndex C))) -' 1 < width (Gauge C,(ApproxIndex C)) by A13, A11, NAT_1:14, XREAL_1:235;
then cell (Gauge C,(ApproxIndex C)),((X-SpanStart C,(ApproxIndex C)) -' 1),((width (Gauge C,(ApproxIndex C))) -' 1) c= UBD C by A6, A8, A14, A9, A5, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A4, A10, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;