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,q being Point of TOP-REAL 2
  st P is_an_arc_of p1,p2 & q <> p2 holds not p2 in L_Segment(P,p1,p2,q)
proof
  let P be Subset of TOP-REAL 2, p1,p2,q be Point of TOP-REAL 2 such that
A1: P is_an_arc_of p1,p2 and
A2: q <> p2;
  assume p2 in L_Segment(P,p1,p2,q);
  then ex w being Point of TOP-REAL 2 st p2 = w & LE w,q,P,p1,p2;
  hence contradiction by A1,A2,Th55;
end;
