
theorem Th1:
  for P being Subset of TOP-REAL 2, p1,p2,p being Point of TOP-REAL
  2 st P is_an_arc_of p1,p2 & p in P holds Segment(P,p1,p2,p,p)={p}
proof
  let P be Subset of TOP-REAL 2,p1,p2,p be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2 and
A2: p in P;
A3: Segment(P,p1,p2,p,p)={q where q is Point of TOP-REAL 2: LE p,q,P,p1,p2 &
  LE q,p,P,p1,p2} by JORDAN6:26;
A4: {p} c= Segment(P,p1,p2,p,p)
  proof
    let x be object;
    assume x in {p};
    then
A5: x=p by TARSKI:def 1;
    LE p,p,P,p1,p2 by A2,JORDAN5C:9;
    hence thesis by A3,A5;
  end;
  Segment(P,p1,p2,p,p)c= {p}
  proof
    let x be object;
    assume x in Segment(P,p1,p2,p,p);
    then consider q being Point of TOP-REAL 2 such that
A6: x=q and
A7: LE p,q,P,p1,p2 & LE q,p,P,p1,p2 by A3;
    p=q by A1,A7,JORDAN5C:12;
    hence thesis by A6,TARSKI:def 1;
  end;
  hence thesis by A4,XBOOLE_0:def 10;
end;
