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 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 holds
  L_Segment(P,p1,p2,p1)={p1}
proof
  let P be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2;
  assume
A1: P is_an_arc_of p1,p2;
  then
A2: p1 in P by TOPREAL1:1;
  thus L_Segment(P,p1,p2,p1) c= {p1}
  proof
    let x be object;
    assume x in L_Segment(P,p1,p2,p1);
    then consider q such that
A3: q=x and
A4: LE q,p1,P,p1,p2;
    q in P by A4;
    then LE p1,q,P,p1,p2 by A1,JORDAN5C:10;
    then q=p1 by A1,A4,JORDAN5C:12;
    hence thesis by A3,TARSKI:def 1;
  end;
  let x be object;
  assume x in {p1};
  then
A5: x=p1 by TARSKI:def 1;
  LE p1,p1,P,p1,p2 by A2,JORDAN5C:9;
  hence thesis by A5;
end;
