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
  (ex i,j st i <= len Gauge(C,n) & j <= width Gauge(C,n) & cell(Gauge(C,
  n),i,j) c= BDD C) implies n >= 1
proof
A1: 2|^0 = 1 by NEWTON:4;
  given i,j such that
A2: i <= len Gauge(C,n) and
A3: j <= width Gauge(C,n) and
A4: cell(Gauge(C,n),i,j) c= BDD C;
A5: j + 1 < width Gauge(C,n) by A2,A3,A4,Th39;
A6: i + 1 < len Gauge(C,n) by A2,A3,A4,Th40;
  assume
A7: n < 1;
  len Gauge(C,n) = 2|^n + 3 by JORDAN8:def 1;
  then
A8: len Gauge(C,n) = 1 + 3 by A1,A7,NAT_1:14;
  width Gauge(C,n) = 2|^n + 3 by JORDAN1A:28;
  then
A9: width Gauge(C,n) = 1 + 3 by A1,A7,NAT_1:14;
  i <= 4 by A8,A2;
  then
A10: i = 0 or ... or i = 4;
  j <= 4 by A3,A9;
  then j = 0 or ... or j = 4;
  then per cases by A10;
  suppose
    j= 0 or j=1;
    hence thesis by A2,A3,A4,Th37;
  end;
  suppose
    i= 0 or i=1;
    hence thesis by A2,A3,A4,Th38;
  end;
  suppose
    j=2 & i=2;
    then cell(Gauge(C,0),2,2) c= BDD C by A4,A7,NAT_1:14;
    hence contradiction by Th18;
  end;
  suppose
    j=3 or j=4 or i=3 or i=4;
    hence thesis by A5,A6,A9,A8;
  end;
end;
