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

theorem
  n is_sufficiently_large_for C implies [X-SpanStart(C,n)-'1,Y-SpanStart
  (C,n)] in Indices Gauge(C,n)
proof
A1: 1 <= X-SpanStart(C,n)-'1 by JORDAN1H:50;
A2: X-SpanStart(C,n)-'1 < len Gauge(C,n) by JORDAN1H:50;
  assume
A3: n is_sufficiently_large_for C;
  then
A4: Y-SpanStart(C,n) <= width Gauge(C,n) by Th7;
  1 < Y-SpanStart(C,n) by A3,Th7;
  hence thesis by A1,A2,A4,MATRIX_0:30;
end;
