reserve j for Nat;

theorem Th24:
  for P being non empty Subset of TOP-REAL 2, p1,p2 being Point of
  TOP-REAL 2 st P is_an_arc_of p1,p2 holds Segment(P,p1,p2,p1,p2)=P
proof
  let P be non empty Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2;
  assume P is_an_arc_of p1,p2;
  then
A1: R_Segment(P,p1,p2,p1)=P & L_Segment(P,p1,p2,p2)=P by JORDAN6:22,24;
  R_Segment(P,p1,p2,p1) /\ L_Segment(P,p1,p2,p2)=Segment(P,p1,p2,p1,p2) by
JORDAN6:def 5;
  hence thesis by A1;
end;
