reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  p in Vertical_Line s & q in Vertical_Line s implies LSeg(p,q) c=
  Vertical_Line s
proof
A1: Vertical_Line(s) = {p1 where p1 is Point of TOP-REAL 2: p1`1=s} by
JORDAN6:def 6;
  assume p in Vertical_Line s & q in Vertical_Line s;
  then
A2: p`1 = s & q`1 = s by JORDAN6:31;
  let u be object;
  assume
A3: u in LSeg(p,q);
  then reconsider p1 = u as Point of TOP-REAL 2;
  p1`1 = s by A2,A3,GOBOARD7:5;
  hence thesis by A1;
end;
