reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 for n,i be Nat st 1 <= i & i <= len Gauge(C,n) & n > 0
  holds Gauge(C,n)*(i,Center Gauge(C,n))`2 = (S-bound C + N-bound C) / 2
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n,i be Nat such that
A1: 1 <= i & i <= len Gauge(C,n);
  len Gauge(C,1) >= 4 by JORDAN8:10;
  then
A2: len Gauge(C,1) >= 1 by XXREAL_0:2;
  assume n > 0;
  hence
  Gauge(C,n)*(i,Center Gauge(C,n))`2 = Gauge(C,1)*(1,Center Gauge(C,1))`2
  by A1,A2,JORDAN1A:37
    .= (S-bound C + N-bound C) / 2 by A2,JORDAN1A:39;
end;
