
theorem
  for p, q being Point of TOP-REAL 2 st p <> q holds LSeg (p, q) =
  { p1 where p1 is Point of TOP-REAL 2 : LE p, p1, p, q & LE p1, q, p, q }
proof
  let p, q be Point of TOP-REAL 2;
  assume
A1: p <> q;
  thus LSeg (p, q) c=
  { p1 where p1 is Point of TOP-REAL 2 : LE p, p1, p, q & LE p1, q, p, q }
  proof
    let a be object;
    assume
A2: a in LSeg (p, q);
    then reconsider a9 = a as Point of TOP-REAL 2;
A3: LE p, a9, p, q by A1,A2,Th10;
    LE a9, q, p, q by A1,A2,Th11;
    hence thesis by A3;
  end;
  thus { p1 where p1 is Point of TOP-REAL 2 :
  LE p, p1, p, q & LE p1, q, p, q } c= LSeg (p, q)
  proof
    let a be object;
    assume a in { p1 where p1 is Point of TOP-REAL 2 :
    LE p, p1, p, q & LE p1, q, p, q };
    then ex a9 be Point of TOP-REAL 2 st ( a9 = a)&( LE p, a9, p, q)
    &( LE a9, q, p, q);
    hence thesis;
  end;
end;
