reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  p in LSeg(p1,p2) & LSeg(p1,p2) is vertical implies LSeg(p,p2) is vertical
proof
  assume
A1: p in LSeg(p1,p2);
  assume
A2: LSeg(p1,p2) is vertical;
  then
A3: p1`1 = p2`1 by SPPOL_1:16;
  p1 in LSeg(p1,p2) by RLTOPSP1:68;
  then p`1 = p1`1 by A1,A2;
  hence thesis by A3,SPPOL_1:16;
end;
