reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem Th43:
  for f being Function of [:S,T:], [:T,S:] st f = <:pr2(the
carrier of S,the carrier of T), pr1(the carrier of S,the carrier of T):> holds
  f is being_homeomorphism
proof
  let f be Function of [:S,T:], [:T,S:] such that
A1: f = <:pr2(the carrier of S,the carrier of T), pr1(the carrier of S,
  the carrier of T):>;
  thus dom f = [#][:S,T:] by FUNCT_2:def 1;
  thus
A2: rng f = [:the carrier of T,the carrier of S:] by A1,Th4
    .= [#][:T,S:] by BORSUK_1:def 2;
  thus
 f is one-to-one by A1;
  thus f is continuous by A1,Th42;
  f is onto by A2,FUNCT_2:def 3;
  then f" = (f qua Function)" by A1,TOPS_2:def 4
    .= <:pr2(the carrier of T,the carrier of S), pr1(the carrier of T,the
  carrier of S):> by A1,Th8;
  hence thesis by Th42;
end;
