reserve i,j,k,n,m for Nat;

theorem
  for p,q,r being Point of TOP-REAL 2 st LSeg(p,q) is vertical & LSeg(q,
  r) is horizontal holds LSeg(p,q) /\ LSeg(q,r) = {q}
proof
  let p,q,r be Point of TOP-REAL 2 such that
A1: LSeg(p,q) is vertical and
A2: LSeg(q,r) is horizontal;
  for x being object holds x in LSeg(p,q) /\ LSeg(q,r) iff x = q
  proof
    let x be object;
    thus x in LSeg(p,q) /\ LSeg(q,r) implies x = q
    proof
      assume
A3:   x in LSeg(p,q) /\ LSeg(q,r);
      then reconsider x as Point of TOP-REAL 2;
      x in LSeg(q,r) by A3,XBOOLE_0:def 4;
      then
A4:   x`2 = q`2 by A2,SPPOL_1:40;
      x in LSeg(p,q) by A3,XBOOLE_0:def 4;
      then x`1 = q`1 by A1,SPPOL_1:41;
      hence thesis by A4,TOPREAL3:6;
    end;
    assume
A5: x = q;
    then
A6: x in LSeg(q,r) by RLTOPSP1:68;
    x in LSeg(p,q) by A5,RLTOPSP1:68;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
