reserve n for Nat;

theorem Th35:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat, i be Nat st 1 <= i & i <= width Gauge(C,n)
& n > 0 holds Gauge(C,n)*(Center Gauge(C,n),i)`1 = (W-bound C + E-bound C) / 2
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat, i be Nat such that
A1: 1 <= i & i <= width Gauge(C,n);
  reconsider ii=i as Nat;
A2: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  assume
A3: n > 0;
  len Gauge(C,1) >= 4 by JORDAN8:10;
  then
A4: len Gauge(C,1) >= 1 by XXREAL_0:2;
  thus Gauge(C,n)*(Center Gauge(C,n),i)`1 = Gauge(C,n)*(Center Gauge(C,n),ii)
  `1
    .= Gauge(C,1)*(Center Gauge(C,1),1)`1 by A1,A2,A4,A3,JORDAN1A:36
    .= (W-bound C + E-bound C) / 2 by A4,JORDAN1A:38;
end;
