reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th42:
  for p1, p2 being Point of TOP-REAL n, P being Subset of TOP-REAL n
  holds P is_an_arc_of p1,p2 implies
  ex F being Path of p1,p2, f being Function of I[01], (TOP-REAL n)|P st
  rng f = P & F = f
proof
  let p1, p2 be Point of TOP-REAL n, P be Subset of TOP-REAL n;
  assume
A1: P is_an_arc_of p1,p2;
  then reconsider P1 = P as non empty Subset of TOP-REAL n by TOPREAL1:1;
  consider h being Function of I[01], (TOP-REAL n)|P such that
A2: h is being_homeomorphism and
A3: h.0 = p1 and
A4: h.1 = p2 by A1,TOPREAL1:def 1;
  h is Function of I[01], (TOP-REAL n)|P1;
  then reconsider h1 = h as Function of I[01],TOP-REAL n by TOPREALA:7;
  h1 is continuous by A2,PRE_TOPC:26;
  then reconsider f = h as Path of p1,p2 by A3,A4,BORSUK_2:def 4;
  take f, h;
  thus rng h = [#]((TOP-REAL n)|P) by A2,TOPS_2:def 5
    .= P by PRE_TOPC:8;
  thus thesis;
end;
