theorem
  1 <= i & i <= len Gauge(C,1) implies Gauge(C,1)*(i,Center Gauge(C,1))
  `2 = (S-bound C + N-bound C) / 2
proof
  set a = N-bound C, s = S-bound C, w = W-bound C, e = E-bound C, G = Gauge(C,
  1);
  assume 1 <= i & i <= len G;
  then [i,Center G] in Indices G by Lm5;
  hence
  G*(i,Center G)`2 = |[w+((e-w)/(2|^1))*(i-2),s+((a-s)/(2|^1))*(Center G-
  2)]|`2 by JORDAN8:def 1
    .= s+(a-s)/(2|^1)*(Center G-2) by EUCLID:52
    .= s+(a-s)/2 by Lm6
    .= (s + a) / 2;
end;
