let n be Nat; :: thesis: for C being being_simple_closed_curve Subset of (TOP-REAL 2) st n is_sufficiently_large_for C holds
[(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n))
let C be being_simple_closed_curve Subset of (TOP-REAL 2); :: thesis: ( n is_sufficiently_large_for C implies [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) )
A1: X-SpanStart (C,n) < len (Gauge (C,n)) by JORDAN1H:49;
1 + 1 < X-SpanStart (C,n) by JORDAN1H:49;
then A2: 1 < X-SpanStart (C,n) by NAT_1:13;
assume A3: n is_sufficiently_large_for C ; :: thesis: [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n))
then A4: Y-SpanStart (C,n) <= width (Gauge (C,n)) by Th7;
1 < Y-SpanStart (C,n) by A3, Th7;
hence [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A2, A1, A4, MATRIX_0:30; :: thesis: verum