
theorem
  for S, T be non empty TopSpace st S,T are_homeomorphic &
  S is pathwise_connected holds T is pathwise_connected
proof
  let S, T be non empty TopSpace;
  given h being Function of S,T such that
A1: h is being_homeomorphism;
  assume
A2: for a, b being Point of S holds a,b are_connected;
  let a, b being Point of T;
  h".a,h".b are_connected by A2;
  then consider f being Function of I[01], S such that
A3: f is continuous and
A4: f.0 = h".a and
A5: f.1 = h".b;
  take g = h*f;
  h is continuous by A1;
  hence g is continuous by A3;
A6: h is one-to-one & rng h = [#]T by A1;
  thus g.0 = h.(h".a) by A4,BORSUK_1:def 14,FUNCT_2:15
    .= (h*h").a by FUNCT_2:15
    .= (id T).a by A6,TOPS_2:52
    .= a;
  thus g.1 = h.(h".b) by A5,BORSUK_1:def 15,FUNCT_2:15
    .= (h*h").b by FUNCT_2:15
    .= (id T).b by A6,TOPS_2:52
    .= b;
end;
