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
  1 <= i & i < len G & 1 <= j & j <= width G & 1 <= m & m <= len G & 1
<= n & n <= width G & p in cell(G,i,j) & p`2 = G*(m,n)`2 implies j = n or j = n
  -' 1
proof
  assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j <= width G and
A5: 1 <= m and
A6: m <= len G and
A7: 1 <= n and
A8: n <= width G and
A9: p in cell(G,i,j) and
A10: p`2 = G*(m,n)`2;
A11: G*(1,n)`2 = G*(m,n)`2 by A5,A6,A7,A8,GOBOARD5:1;
A12: 1 <= len G by A1,A2,XXREAL_0:2;
  per cases by A4,XXREAL_0:1;
  suppose
    j = width G;
    hence thesis by A1,A2,A5,A6,A7,A8,A9,A10,Th23;
  end;
  suppose
    j < width G;
    then
    cell(G,i,j) = { |[r,s]| where r, s is Real:
G*(i,1)`1 <= r & r <= G*(
    i+1,1)`1 & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 } by A1,A2,A3,GOBRD11:32;
    then consider r, s being Real such that
A13: p = |[r,s]| and
    G*(i,1)`1 <= r and
    r <= G*(i+1,1)`1 and
A14: G*(1,j)`2 <= s and
A15: s <= G*(1,j+1)`2 by A9;
A16: p`2 = s by A13,EUCLID:52;
    j <= n & n <= j+1
    proof
      assume
A17:  not thesis;
      per cases by A17;
      suppose
        j > n;
        hence contradiction by A4,A7,A10,A12,A11,A14,A16,GOBOARD5:4;
      end;
      suppose
A18:    n > j+1;
        1 <= j+1 by NAT_1:11;
        hence contradiction by A8,A10,A12,A11,A15,A16,A18,GOBOARD5:4;
      end;
    end;
    hence thesis by Lm2;
  end;
end;
