
theorem Th15:
  for S,T being non empty Poset holds S,T are_isomorphic iff ex f
being monotone Function of S,T, g being monotone Function of T,S st f*g = id T
  & g*f = id S
proof
  let S,T be non empty Poset;
  hereby
    assume S,T are_isomorphic;
    then consider f being Function of S,T such that
A1: f is isomorphic;
    reconsider f as monotone Function of S,T by A1;
    consider g being Function of T,S such that
A2: g = f qua Function" and
A3: g is monotone by A1,WAYBEL_0:def 38;
    take f;
    reconsider g as monotone Function of T,S by A3;
    take g;
    rng f = the carrier of T by A1,WAYBEL_0:66;
    hence f*g = id T & g*f = id S by A1,A2,FUNCT_2:29;
  end;
  given f being monotone Function of S,T, g being monotone Function of T,S
  such that
A4: f*g = id T and
A5: g*f = id S;
  take f;
A6: f is one-to-one by A5,FUNCT_2:23;
  f is onto by A4,FUNCT_2:23;
  then rng f = the carrier of T by FUNCT_2:def 3;
  then g = f qua Function" by A5,A6,FUNCT_2:30;
  hence thesis by A6,WAYBEL_0:def 38;
end;
