reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  1 <= i & i <= len Gauge (D,n) & 1 <= j & j <= len Gauge (D,m) & (n > 0
& m > 0 or n = 0 & m = 0) implies Gauge(D,n)*(i, Center Gauge(D,n))`2 = Gauge(D
  ,m)*(j, Center Gauge(D,m))`2
proof
  set a = N-bound D, s = S-bound D, w = W-bound D, e = E-bound D, G = Gauge(D,
  n), M = Gauge(D,m);
  assume 1 <= i & i <= len G;
  then
A1: [i,Center G] in Indices G by Lm5;
  assume 1 <= j & j <= len M;
  then
A2: [j,Center M] in Indices M by Lm5;
  assume
A3: n > 0 & m > 0 or n = 0 & m = 0;
  per cases by A3;
  suppose that
A4: n > 0 and
A5: m > 0;
    thus G*(i,Center G)`2 = |[w+(e-w)/(2|^n)*(i - 2), s+(a-s)/(2|^n)*(Center G
    - 2)]|`2 by A1,JORDAN8:def 1
      .= s+(a-s)/(2|^n)*(Center G - 2) by EUCLID:52
      .= s+(a-s)/2 by A4,Lm6
      .= s+(a-s)/(2|^m)*(Center M - 2) by A5,Lm6
      .= |[w+(e-w)/(2|^m)*(j - 2), s+(a-s)/(2|^m)*(Center M - 2)]|`2 by
EUCLID:52
      .= M*(j,Center M)`2 by A2,JORDAN8:def 1;
  end;
  suppose
A6: n = 0 & m = 0;
    thus G*(i,Center G)`2 = |[w+(e-w)/(2|^n)*(i - 2), s+(a-s)/(2|^n)*(Center G
    - 2)]|`2 by A1,JORDAN8:def 1
      .= s+(a-s)/(2|^m)*(Center M - 2) by A6,EUCLID:52
      .= |[w+(e-w)/(2|^m)*(j - 2), s+(a-s)/(2|^m)*(Center M - 2)]|`2 by
EUCLID:52
      .= M*(j,Center M)`2 by A2,JORDAN8:def 1;
  end;
end;
