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+1 <= len G & 1 <=
j1 & j1 <= width G & 1 <= i2 & i2+1 <= len G & 1 <= j2 & j2 <= width G & LSeg(G
*(i1,j1),G*(i1+1,j1)) meets LSeg(G*(i2,j2),G*(i2+1,j2)) holds i1 = i2 & LSeg(G*
(i1,j1),G*(i1+1,j1)) = LSeg(G*(i2,j2),G*(i2+1,j2)) or i1 = i2+1 & LSeg(G*(i1,j1
  ),G*(i1+1,j1)) /\ LSeg(G*(i2,j2),G*(i2+1,j2)) = { G* (i1,j1) } or i1+1 = i2 &
  LSeg(G*(i1,j1),G*(i1+1,j1)) /\ LSeg(G*(i2,j2),G*(i2+1,j2)) = { G* (i2,j2) }
proof
  let i1,j1,i2,j2 be Nat such that
A1: 1 <= i1 and
A2: i1+1 <= len G and
A3: 1 <= j1 & j1 <= width G and
A4: 1 <= i2 and
A5: i2+1 <= len G and
A6: 1 <= j2 & j2 <= width G & LSeg(G*(i1,j1),G*(i1+1,j1)) meets LSeg(G*(
  i2, j2),G*(i2+1,j2));
A7: j1 = j2 by A1,A2,A3,A4,A5,A6,Th20;
A8: i1+1+1 = i1+(1+1);
A9: i2+1+1 = i2+(1+1);
A10: |.i1-i2.| = 0 or |.i1-i2.| = 1 by A1,A2,A3,A4,A5,A6,Th20,NAT_1:25;
  per cases by A10,Th2,SEQM_3:41;
  case
    i1 = i2;
    hence thesis by A7;
  end;
  case
    i1 = i2+1;
    hence thesis by A2,A3,A4,A7,A9,Th14;
  end;
  case
    i1+1 = i2;
    hence thesis by A1,A3,A5,A7,A8,Th14;
  end;
end;
