reserve C, P for Simple_closed_curve,
  a, b, c, d, e for Point of TOP-REAL 2;

theorem Th1:
  for n being Element of NAT, a, p1, p2 being Point of TOP-REAL n,
  P being Subset of TOP-REAL n st a in P & P is_an_arc_of p1,p2 ex f being
Function of I[01], (TOP-REAL n)|P, r being Real st f is being_homeomorphism & f
  .0 = p1 & f.1 = p2 & 0 <= r & r <= 1 & f.r = a
proof
  let n be Element of NAT, a, p1, p2 be Point of TOP-REAL n, P be Subset of
  TOP-REAL n such that
A1: a in P;
  given f being Function of I[01], (TOP-REAL n)|P such that
A2: f is being_homeomorphism and
A3: f.0 = p1 & f.1 = p2;
  rng f = [#]((TOP-REAL n)|P) by A2,TOPS_2:def 5
    .= the carrier of (TOP-REAL n)|P
    .= P by PRE_TOPC:8;
  then consider r being object such that
A4: r in dom f and
A5: a = f.r by A1,FUNCT_1:def 3;
A6: dom f = [#]I[01] by A2,TOPS_2:def 5
    .= [.0,1.] by BORSUK_1:40;
  then reconsider r as Real by A4;
  take f, r;
  thus f is being_homeomorphism & f.0 = p1 & f.1 = p2 by A2,A3;
  thus 0 <= r & r <= 1 by A4,A6,XXREAL_1:1;
  thus thesis by A5;
end;
