reserve i,j,k,n for Nat,
  C for being_simple_closed_curve Subset of TOP-REAL 2;

theorem Th2:
  Y-InitStart C > 1
proof
  set m = ApproxIndex C;
A1: X-SpanStart(C,m)-'1 <= len Gauge(C,m) by JORDAN1H:50;
  assume
A2: Y-InitStart C <= 1;
  per cases by A2,NAT_1:25;
  suppose
A3: Y-InitStart C = 0;
A4: cell(Gauge(C,m),X-SpanStart(C,m)-'1,0) c= UBD C by A1,JORDAN1A:49;
    0 <= width Gauge(C,m);
    then
A5: cell(Gauge(C,m),X-SpanStart(C,m)-'1,0) is non empty by A1,JORDAN1A:24;
    cell(Gauge(C,m),X-SpanStart(C,m)-'1,0) c= BDD C by A3,Def2;
    hence contradiction by A5,A4,JORDAN2C:24,XBOOLE_1:68;
  end;
  suppose
    Y-InitStart C = 1;
    then
A6: cell(Gauge(C,m),X-SpanStart(C,m)-'1,1) c= BDD C by Def2;
    set i1 = X-SpanStart(C,m);
A7: i1-'1 <= i1 by NAT_D:44;
    i1 < len Gauge(C,m) by JORDAN1H:49;
    then
A8: i1-'1 < len Gauge(C,m) by A7,XXREAL_0:2;
    BDD C c= C` by JORDAN2C:25;
    then
A9: cell(Gauge(C,m),X-SpanStart(C,m)-'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,m) <> 0 by MATRIX_0:def 10;
    then
A12: 0+1 <= width Gauge(C,m) by NAT_1:14;
    then
A13: cell(Gauge(C,m),X-SpanStart(C,m)-'1,1) is non empty by A1,JORDAN1A:24;
    1 <= i1-'1 by JORDAN1H:50;
    then
    cell(Gauge(C,m),i1-'1,0) /\ cell(Gauge(C,m),i1-'1,0+1) = LSeg(Gauge(C
    ,m)*(i1-'1,0+1),Gauge(C,m)*(i1-'1+1,0+1)) by A11,A8,GOBOARD5:26;
    then
A14: cell(Gauge(C,m),i1-'1,0) meets cell(Gauge(C,m),i1-'1,0+1);
    cell(Gauge(C,m),X-SpanStart(C,m)-'1,0) c= UBD C by A1,JORDAN1A:49;
    then cell(Gauge(C,m),X-SpanStart(C,m)-'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;
  end;
end;
