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

theorem Th4:
  n is_sufficiently_large_for C implies X-SpanStart(C,n) = 2|^(n-'
  ApproxIndex C)*(X-SpanStart(C,ApproxIndex C)-2)+2
proof
  set m = ApproxIndex C;
A1: m >= 1 by Th1;
  assume n is_sufficiently_large_for C;
  then
A2: n >=ApproxIndex C by Def1;
  (n-'m)+(m-'1) = (n-'m)+(m-1) by Th1,XREAL_1:233
    .= (n-'m)+ m-1
    .= n-1 by A2,XREAL_1:235
    .= n-'1 by A2,A1,XREAL_1:233,XXREAL_0:2;
  hence thesis by NEWTON:8;
end;
