reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th1:
  for P being Subset of T, p1,p2 being Point of T st P is_an_arc_of
  p1,p2 holds p1 in P & p2 in P
proof
  let P be Subset of T, p1,p2 be Point of T;
  assume P is_an_arc_of p1,p2;
  then consider f being Function of I[01], T|P such that
A1: f is being_homeomorphism and
A2: f.0 = p1 and
A3: f.1 = p2;
A4: dom f = [#]I[01] by A1,TOPS_2:def 5
    .= the carrier of I[01];
  1 in [.0,1.] by XXREAL_1:1;
  then
A5: p2 in rng f by A3,A4,BORSUK_1:40,FUNCT_1:def 3;
A6: rng f = [#](T|P) by A1,TOPS_2:def 5;
  0 in [.0,1.] by XXREAL_1:1;
  then p1 in rng f by A2,A4,BORSUK_1:40,FUNCT_1:def 3;
  hence thesis by A5,A6,PRE_TOPC:def 5;
end;
