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 <= len G & 1 <= j2 & j2+1 <= width G & LSeg
(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2,j2+1)) holds j1 = j2 & LSeg(
G*(i1,j1),G*(i1,j1+1)) = LSeg(G*(i2,j2),G*(i2,j2+1)) or j1 = j2+1 & LSeg(G*(i1,
j1),G*(i1,j1+1)) /\ LSeg(G*(i2,j2),G*(i2,j2+1)) = { G* (i1,j1) } or j1+1 = j2 &
  LSeg(G*(i1,j1),G*(i1,j1+1)) /\ LSeg(G*(i2,j2),G*(i2,j2+1)) = { G* (i2,j2) }
proof
  let i1,j1,i2,j2 be Nat such that
A1: 1 <= i1 & i1 <= len G and
A2: 1 <= j1 and
A3: j1+1 <= width G and
A4: 1 <= i2 & i2 <= len G and
A5: 1 <= j2 and
A6: j2+1 <= width G and
A7: LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2,j2+1));
A8: i1 = i2 by A1,A2,A3,A4,A5,A6,A7,Th19;
A9: j1+1+1 = j1+(1+1);
A10: j2+1+1 = j2+(1+1);
A11: |.j1-j2.| = 0 or |.j1-j2.| = 1 by A1,A2,A3,A4,A5,A6,A7,Th19,NAT_1:25;
  per cases by A11,Th2,SEQM_3:41;
  case
    j1 = j2;
    hence thesis by A8;
  end;
  case
    j1 = j2+1;
    hence thesis by A1,A3,A5,A8,A10,Th13;
  end;
  case
    j1+1 = j2;
    hence thesis by A1,A2,A6,A8,A9,Th13;
  end;
end;
