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

theorem Th7:
  n is_sufficiently_large_for C implies 1 < Y-SpanStart(C,n) &
  Y-SpanStart(C,n) <= width Gauge(C,n)
proof
  assume
A1: n is_sufficiently_large_for C;
  thus 1 < Y-SpanStart(C,n)
  proof
A2: X-SpanStart(C,n)-'1 <= len Gauge(C,n) by JORDAN1H:50;
    assume
A3: Y-SpanStart(C,n) <= 1;
    per cases by A3,NAT_1:25;
    suppose
A4:   Y-SpanStart(C,n) = 0;
A5:   cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) c= UBD C by A2,JORDAN1A:49;
      0 <= width Gauge(C,n);
      then
A6:   cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) is non empty by A2,JORDAN1A:24;
      cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) c= BDD C by A1,A4,Th6;
      hence contradiction by A6,A5,JORDAN2C:24,XBOOLE_1:68;
    end;
    suppose
      Y-SpanStart(C,n) = 1;
      then
A7:   cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) c= BDD C by A1,Th6;
      set i1 = X-SpanStart(C,n);
A8:   i1-'1 <= i1 by NAT_D:44;
      i1 < len Gauge(C,n) by JORDAN1H:49;
      then
A9:   i1-'1 < len Gauge(C,n) by A8,XXREAL_0:2;
      BDD C c= C` by JORDAN2C:25;
      then
A10:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) c= C` by A7;
      UBD C is_outside_component_of C by JORDAN2C:68;
      then
A11:  UBD C is_a_component_of C` by JORDAN2C:def 3;
A12:  width Gauge(C,n) <> 0 by MATRIX_0:def 10;
      then
A13:  0+1 <= width Gauge(C,n) by NAT_1:14;
      then
A14:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) is non empty by A2,JORDAN1A:24;
      1 <= i1-'1 by JORDAN1H:50;
      then
      cell(Gauge(C,n),i1-'1,0) /\ cell(Gauge(C,n),i1-'1,0+1) = LSeg(Gauge
      (C,n)*(i1-'1,0+1),Gauge(C,n)*(i1-'1+1,0+1)) by A12,A9,GOBOARD5:26;
      then
A15:  cell(Gauge(C,n),i1-'1,0) meets cell(Gauge(C,n),i1-'1,0+1);
      cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) c= UBD C by A2,JORDAN1A:49;
      then cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) c= UBD C by A13,A9,A15,A11
,A10,GOBOARD9:4,JORDAN1A:25;
      hence contradiction by A7,A14,JORDAN2C:24,XBOOLE_1:68;
    end;
  end;
  thus thesis by A1,Def3;
end;
