
theorem Th7:
  for P, Q being Subset of TOP-REAL 2, p1, p2 being Point of
TOP-REAL 2 st P c= Q & P is closed & P is_an_arc_of p1, p2 holds First_Point (P
  , p1, p2, Q) = p1 & Last_Point (P, p1, p2, Q) = p2
proof
  let P, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2;
  assume that
A1: P c= Q and
A2: P is closed and
A3: P is_an_arc_of p1, p2;
A4: p2 in P by A3,TOPREAL1:1;
  P /\ Q = P & p1 in P by A1,A3,TOPREAL1:1,XBOOLE_1:28;
  hence thesis by A1,A2,A3,A4,Th3,Th6;
end;
