reserve p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for P being compact non empty Subset of TOP-REAL 2, q being Point of
  TOP-REAL 2 st P is being_simple_closed_curve & q in P & q<>W-min(P) holds
  Segment(q,q,P)={q}
proof
  let P be compact non empty Subset of TOP-REAL 2, q be Point of TOP-REAL 2;
  assume that
A1: P is being_simple_closed_curve and
A2: q in P and
A3: q <> W-min P;
  for x being object holds x in Segment(q,q,P) iff x=q
  proof
    let x be object;
    hereby
      assume x in Segment(q,q,P);
      then x in {p: LE q,p,P & LE p,q,P} by A3,Def1;
      then ex p st p=x & LE q,p,P & LE p,q,P;
      hence x=q by A1,JORDAN6:57;
    end;
    assume
A4: x=q;
    LE q,q,P by A1,A2,JORDAN6:56;
    then x in {p: LE q,p,P & LE p,q,P} by A4;
    hence thesis by A3,Def1;
  end;
  hence thesis by TARSKI:def 1;
end;
