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 Th8:
  p`1 <= q`1 & q`1 <= p1`1 & p`2 = q`2 & q`2 = p1`2 implies q in LSeg(p,p1)
proof
  assume that
A1: p`1 <= q`1 & q`1 <= p1`1 and
A2: p`2 = q`2 and
A3: q`2 = p1`2;
A4: p`1 <= p1`1 by A1,XXREAL_0:2;
  per cases by A4,XXREAL_0:1;
  suppose
A5: p`1 = p1`1;
    then p`1 = q`1 by A1,XXREAL_0:1;
    then
A6: q = |[p`1,p`2]| by A2,EUCLID:53
      .= p by EUCLID:53;
    p = |[p1`1,p1`2]| by A2,A3,A5,EUCLID:53
      .= p1 by EUCLID:53;
    then LSeg(p,p1) = {p} by RLTOPSP1:70;
    hence thesis by A6,TARSKI:def 1;
  end;
  suppose
A7: p`1 < p1`1;
A8: q in {q1: q1`2 = q`2 & p`1 <= q1`1 & q1`1 <= p1`1} by A1;
    p = |[p`1,q`2]| & p1 = |[p1`1,q`2]| by A2,A3,EUCLID:53;
    hence thesis by A7,A8,TOPREAL3:10;
  end;
end;
