reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;

theorem
  for i1,j1,i2,j2 being Nat st 1 <= i1 & i1 <= len G & 1 <=
j1 & j1+1 <= width G & 1 <= i2 & i2+1 <= len G & 1 <= j2 & j2 <= width G & LSeg
(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2+1,j2)) holds j1 = j2 & LSeg(
G*(i1,j1),G*(i1,j1+1)) /\ LSeg(G*(i2,j2),G*(i2+1,j2)) = { G* (i1,j1) } or j1+1
= j2 & LSeg(G*(i1,j1),G*(i1,j1+1)) /\ LSeg(G*(i2,j2),G*(i2+1,j2)) = { G* (i1,j1
  +1) }
proof
  let i1,j1,i2,j2 be Nat such that
A1: 1 <= i1 & i1 <= len G and
A2: 1 <= j1 & j1+1 <= width G & 1 <= i2 & i2+1 <= len G and
A3: 1 <= j2 & j2 <= width G & LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(
  i2, j2),G*(i2+1,j2));
  per cases by A1,A2,A3,Th21;
  case
A4: j1 = j2;
    now
      per cases by A1,A2,A3,Th21;
      suppose
        i1 = i2;
        hence thesis by A2,A4,Th17;
      end;
      suppose
        i1 = i2+1;
        hence thesis by A2,A4,Th18;
      end;
    end;
    hence thesis;
  end;
  case
A5: j1+1 = j2;
    now
      per cases by A1,A2,A3,Th21;
      suppose
        i1 = i2;
        hence thesis by A2,A5,Th15;
      end;
      suppose
        i1 = i2+1;
        hence thesis by A2,A5,Th16;
      end;
    end;
    hence thesis;
  end;
end;
