
theorem Th1:
  for n being Nat, p, q being Point of TOP-REAL n, P
  being Subset of TOP-REAL n st P is_an_arc_of p, q holds P is compact
proof
  let n be Nat;
  let p, q be Point of TOP-REAL n;
  let P be Subset of TOP-REAL n;
  assume P is_an_arc_of p, q;
  then consider f be Function of I[01], (TOP-REAL n)|P such that
A1: f is being_homeomorphism and
  f.0 = p and
  f.1 = q by TOPREAL1:def 1;
  per cases;
  suppose
    P <> {};
    then reconsider P9 = P as non empty Subset of TOP-REAL n;
    f is continuous & rng f = [#] ((TOP-REAL n)|P9) by A1,TOPS_2:def 5;
    then (TOP-REAL n)|P9 is compact by COMPTS_1:14;
    hence thesis by COMPTS_1:3;
  end;
  suppose
    P = {}TOP-REAL n;
    hence thesis;
  end;
end;
