reserve 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,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  Center Gauge(E,n) = 2|^(n-'1) + 2
proof
   reconsider n as Nat;
  set G = Gauge(E,n);
  per cases;
  suppose
A1: n = 0;
A2: 4 = 2 * 2 + 0;
A3: 0-1 < 0;
    Center G = len G div 2 + 1 by JORDAN1A:def 1;
    then Center G = (2|^0 + 3) div 2 + 1 by A1,JORDAN8:def 1
      .= (1+3) div 2 + 1 by NEWTON:4
      .= 1 + 1 + 1 by A2,NAT_D:def 1
      .= 2|^0 + 1 + 1 by NEWTON:4
      .= 2|^(0-'1) + 1 + 1 by A3,XREAL_0:def 2
      .= 2|^(n-'1) + 2 by A1;
   hence thesis;
  end;
  suppose
A4: n > 0;
    then
A5: 0+1 <= n by NAT_1:13;
A6: 2|^n div 2 = 2|^n / 2 by A4,PEPIN:64
      .= 2|^n / 2|^1
      .= 2|^(n-'1) by A5,TOPREAL6:10;
    3 = 2 * 1 + 1;
    then
A7: 3 div 2 = 1 by NAT_D:def 1;
A8: 2|^n mod 2 = 0 by A4,PEPIN:36;
    len G div 2 + 1 = (2|^n + 3) div 2 + 1 by JORDAN8:def 1
      .= (2|^n div 2) + (3 div 2) + 1 by A8,NAT_D:19
      .= (2|^n div 2) + (1 + 1) by A7;
    hence thesis by A6,JORDAN1A:def 1;
  end;
end;
