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`1 = G*(m,n)`1 implies i = m or i = m
  -' 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`1 = G*(m,n)`1;
A11: G*(m,1)`1 = G*(m,n)`1 by A5,A6,A7,A8,GOBOARD5:2;
A12: 1 <= width G by A3,A4,XXREAL_0:2;
  per cases by A2,XXREAL_0:1;
  suppose
    i = len G;
    hence thesis by A3,A4,A5,A6,A7,A8,A9,A10,Th21;
  end;
  suppose
    i < len 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,A3,A4,GOBRD11:32;
    then consider r, s being Real such that
A13: p = |[r,s]| and
A14: G*(i,1)`1 <= r and
A15: r <= G*(i+1,1)`1 and
    G*(1,j)`2 <= s and
    s <= G*(1,j+1)`2 by A9;
A16: p`1 = r by A13,EUCLID:52;
    i <= m & m <= i+1
    proof
      assume
A17:  not thesis;
      per cases by A17;
      suppose
        i > m;
        hence contradiction by A2,A5,A10,A12,A11,A14,A16,GOBOARD5:3;
      end;
      suppose
A18:    m > i+1;
        1 <= i+1 by NAT_1:11;
        hence contradiction by A6,A10,A12,A11,A15,A16,A18,GOBOARD5:3;
      end;
    end;
    hence thesis by Lm2;
  end;
end;
