theorem Th25:
  for p,p1,p2 st not p in LSeg(p1,p2) & p1`2 = p2`2 & p2`2 = p`2
  holds p1 in LSeg(p,p2) or p2 in LSeg(p,p1)
proof
  let p,p1,p2 such that
A1: not p in LSeg(p1,p2) and
A2: p1`2 = p2`2 & p2`2 = p`2;
  per cases;
  suppose
A3: p1`1 <= p2`1;
    now
      per cases by A1,A2,GOBOARD7:8;
      suppose
        p`1<p1`1;
        hence thesis by A2,A3,GOBOARD7:8;
      end;
      suppose
        p2`1<p`1;
        hence thesis by A2,A3,GOBOARD7:8;
      end;
    end;
    hence thesis;
  end;
  suppose
A4: p2`1 <= p1`1;
    now
      per cases by A1,A2,GOBOARD7:8;
      suppose
        p`1<p2`1;
        hence thesis by A2,A4,GOBOARD7:8;
      end;
      suppose
        p1`1<p`1;
        hence thesis by A2,A4,GOBOARD7:8;
      end;
    end;
    hence thesis;
  end;
end;
