
theorem Th14:
  for S,T being non empty reflexive transitive RelStr for f being
  Function of S,T holds f is isomorphic iff f is monotone & ex g being monotone
  Function of T,S st f*g = id T & g*f = id S
proof
  let S,T be non empty reflexive transitive RelStr, f be Function of S,T;
  hereby
    assume
A1: f is isomorphic;
    hence f is monotone;
    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;
    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;
  assume
A4: f is monotone;
  given g being monotone Function of T,S such that
A5: f*g = id T and
A6: g*f = id S;
A7: f is one-to-one by A6,FUNCT_2:23;
  f is onto by A5,FUNCT_2:23;
  then rng f = the carrier of T by FUNCT_2:def 3;
  then g = f qua Function" by A6,A7,FUNCT_2:30;
  hence thesis by A4,A7,WAYBEL_0:def 38;
end;
