
theorem
  for n being Nat, P being Subset of TOP-REAL n,
  p1, p2 being Point of TOP-REAL n st
  P is_an_arc_of p1,p2 holds P is_an_arc_of p2,p1
proof
  let n be Nat, P be Subset of TOP-REAL n,
  p1,p2 be Point of TOP-REAL n;
  assume
A1: P is_an_arc_of p1, p2;
  then consider f being Function of I[01], (TOP-REAL n) | P such that
A2: f is being_homeomorphism and
A3: f.0 = p1 and
A4: f.1 = p2 by TOPREAL1:def 1;
  set Ex = L[01]((0,1)(#),(#)(0,1));
A5: Ex is being_homeomorphism by TREAL_1:18;
  set g = f * Ex;
A6: Ex.(0,1)(#) = 0 by BORSUK_1:def 14,TREAL_1:5,9;
A7: Ex.(#)(0,1) = 1 by BORSUK_1:def 15,TREAL_1:5,9;
  dom f = [#]I[01] by A2,TOPS_2:def 5;
  then rng Ex = dom f by A5,TOPMETR:20,TOPS_2:def 5;
  then
A8: dom g = dom Ex by RELAT_1:27;
  reconsider P as non empty Subset of TOP-REAL n by A1,TOPREAL1:1;
A9: dom g = [#]I[01] by A5,A8,TOPMETR:20,TOPS_2:def 5;
  reconsider g as Function of I[01], (TOP-REAL n) | P by TOPMETR:20;
A10: g is being_homeomorphism by A2,A5,TOPMETR:20,TOPS_2:57;
A11: g.0 = p2 by A4,A7,A9,BORSUK_1:def 14,FUNCT_1:12,TREAL_1:5;
  g.1 = p1 by A3,A6,A9,BORSUK_1:def 15,FUNCT_1:12,TREAL_1:5;
  hence thesis by A10,A11,TOPREAL1:def 1;
end;
