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

theorem
  for p,q,r,s being Point of TOP-REAL 2 st LSeg(p,q) is horizontal &
  LSeg(s,r) is vertical & r in LSeg(p,q) holds LSeg(p,q) /\ LSeg(s,r) = {r}
proof
  let p,q,r,s be Point of TOP-REAL 2 such that
A1: LSeg(p,q) is horizontal and
A2: LSeg(s,r) is vertical and
A3: r in LSeg(p,q);
  for x being object holds x in LSeg(p,q) /\ LSeg(s,r) iff x = r
  proof
    let x be object;
    thus x in LSeg(p,q) /\ LSeg(s,r) implies x = r
    proof
      assume
A4:   x in LSeg(p,q) /\ LSeg(s,r);
      then reconsider x as Point of TOP-REAL 2;
      x in LSeg(p,q) by A4,XBOOLE_0:def 4;
      then
A5:   x`2 = p`2 by A1,SPPOL_1:40;
      x in LSeg(s,r) by A4,XBOOLE_0:def 4;
      then
A6:   x`1 = r`1 by A2,SPPOL_1:41;
      p`2 = r`2 by A1,A3,SPPOL_1:40;
      hence thesis by A5,A6,TOPREAL3:6;
    end;
    assume
A7: x = r;
    then x in LSeg(s,r) by RLTOPSP1:68;
    hence thesis by A3,A7,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
