theorem
  for A being Subset of TOP-REAL 2 st A c= Vertical_Line s holds A is vertical
proof
A1: Vertical_Line(s) = {p where p is Point of TOP-REAL 2: p`1=s} by
JORDAN6:def 6;
  let A being Subset of TOP-REAL 2 such that
A2: A c= Vertical_Line s;
  for p,q being Point of TOP-REAL 2 st p in A & q in A holds p`1=q`1
  proof
    let p,q be Point of TOP-REAL 2;
    assume p in A;
    then p in Vertical_Line s by A2;
    then
A3: ex p1 being Point of TOP-REAL 2 st p1 = p & p1`1 =s by A1;
    assume q in A;
    then q in Vertical_Line s by A2;
    then ex p1 being Point of TOP-REAL 2 st p1 = q & p1`1 =s by A1;
    hence thesis by A3;
  end;
  hence thesis by SPPOL_1:def 3;
end;
