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 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2
  holds
  R_Segment(P,p1,p2,q1)=L_Segment(P,p2,p1,q1)
proof
  let P be Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2;
A2: {q: LE q1,q,P,p1,p2} c= {q2 : LE q2,q1,P,p2,p1}
  proof
    let x be object;
    assume x in {q: LE q1,q,P,p1,p2};
    then consider q such that
A3: q=x and
A4: LE q1,q,P,p1,p2;
    LE q,q1,P,p2,p1 by A1,A4,Th18;
    hence thesis by A3;
  end;
  {q2 : LE q2,q1,P,p2,p1} c= {q: LE q1,q,P,p1,p2}
  proof
    let x be object;
    assume x in {q: LE q,q1,P,p2,p1};
    then consider q such that
A5: q=x and
A6: LE q,q1,P,p2,p1;
    LE q1,q,P,p1,p2 by A1,A6,Th18,JORDAN5B:14;
    hence thesis by A5;
  end;
  hence thesis by A2;
end;
