reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem
  for S, T being being_simple_closed_curve SubSpace of TOP-REAL 2 holds
  S,T are_homeomorphic
proof
  let S, T be being_simple_closed_curve SubSpace of TOP-REAL 2;
  the TopStruct of S, the TopStruct of T are_homeomorphic
  proof
    reconsider A = the carrier of the TopStruct of S as Simple_closed_curve by
Def5;
    consider f being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|A
    such that
A1: f is being_homeomorphism by TOPREAL2:def 1;
A2: f" is being_homeomorphism by A1,TOPS_2:56;
A3: [#]the TopStruct of S = A;
    the TopStruct of S is strict SubSpace of TOP-REAL 2 by TMAP_1:6;
    then
A4: the TopStruct of S = (TOP-REAL 2)|A by A3,PRE_TOPC:def 5;
    reconsider B = the carrier of the TopStruct of T as Simple_closed_curve by
Def5;
    consider g being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|B
    such that
A5: g is being_homeomorphism by TOPREAL2:def 1;
A6: [#]the TopStruct of T = B;
A7: the TopStruct of T is strict SubSpace of TOP-REAL 2 by TMAP_1:6;
    then reconsider
    h = g*f" as Function of the TopStruct of S, the TopStruct of T
    by A4,A6,PRE_TOPC:def 5;
    take h;
    the TopStruct of T = (TOP-REAL 2)|B by A7,A6,PRE_TOPC:def 5;
    hence thesis by A5,A4,A2,TOPS_2:57;
  end;
  hence thesis by TOPREALA:15;
end;
