
theorem
  for S, T being TopStruct st (S,T are_homeomorphic or ex f being
Function of S,T st dom f = [#]S & rng f = [#]T) holds S is empty iff T is empty
proof
  let S, T be TopStruct;
  assume
A1: S,T are_homeomorphic or ex f being Function of S,T st dom f = [#]S &
  rng f = [#]T;
  per cases by A1;
  suppose
    S,T are_homeomorphic;
    then consider f being Function of S, T such that
A2: f is being_homeomorphism;
    rng f = [#]T & dom f = [#]S by A2;
    hence thesis;
  end;
  suppose
    ex f being Function of S,T st dom f = [#]S & rng f = [#]T;
    hence thesis;
  end;
end;
