theorem
  1 <= i & i <= len Gauge(C,1) implies Gauge(C,1)*(Center Gauge(C,1),i)
  `1 = (W-bound C + E-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 [Center G,i] in Indices G by Lm4;
  hence
  G*(Center G,i)`1 = |[w+((e-w)/(2|^1))*(Center G-2),s+((a-s)/(2|^1))*(i-
  2)]|`1 by JORDAN8:def 1
    .= w+(e-w)/(2|^1)*(Center G-2) by EUCLID:52
    .= w+(e-w)/2 by Lm6
    .= (w + e) / 2;
end;
