reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for P being Subset of TOP-REAL 2, p1,p2,q1,q2 being Point of TOP-REAL 2
  holds Segment(P,p1,p2,q1,q2)={q:LE q1,q,P,p1,p2 & LE q,q2,P,p1,p2}
proof
  let P be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2;
  thus Segment(P,p1,p2,q1,q2) c= {q:LE q1,q,P,p1,p2 & LE q,q2,P,p1,p2}
  proof
    let x be object;
    assume
A1: x in Segment(P,p1,p2,q1,q2);
    then
A2: x in R_Segment(P,p1,p2,q1) by XBOOLE_0:def 4;
A3: x in L_Segment(P,p1,p2,q2) by A1,XBOOLE_0:def 4;
A4: ex q st ( q=x)&( LE q1,q,P,p1,p2) by A2;
    ex p st ( p=x)&( LE p,q2,P,p1,p2) by A3;
    hence thesis by A4;
  end;
  let x be object;
  assume x in {q:LE q1,q,P,p1,p2 & LE q,q2,P,p1,p2};
  then
A5: ex q st ( q=x)&( LE q1,q,P,p1,p2)&( LE q,q2,P,p1,p2);
  then
A6: x in R_Segment(P,p1,p2,q1);
  x in L_Segment(P,p1,p2,q2) by A5;
  hence thesis by A6,XBOOLE_0:def 4;
end;
