reserve j for Nat;

theorem
  for P being Subset of TOP-REAL 2, p1,p2,p being Point of TOP-REAL 2,e
being Real st P is_an_arc_of p1,p2 & p1`1<=e & p2`1>=e holds ex p3 being Point
  of TOP-REAL 2 st p3 in P & p3`1=e
proof
  let P be Subset of TOP-REAL 2, p1,p2,p be Point of TOP-REAL 2,e be Real;
  set x = the Element of P /\ Vertical_Line(e);
  assume P is_an_arc_of p1,p2 & p1`1<=e & p2`1>=e;
  then P meets Vertical_Line(e) by JORDAN6:49;
  then
A1: P /\ Vertical_Line(e) <> {} by XBOOLE_0:def 7;
  then x in Vertical_Line(e) by XBOOLE_0:def 4;
  then x in {p3 where p3 is Point of TOP-REAL 2: p3`1=e} by JORDAN6:def 6;
  then
A2: ex p4 being Point of TOP-REAL 2 st p4=x & p4`1=e;
  x in P by A1,XBOOLE_0:def 4;
  hence thesis by A2;
end;
