reserve x, y for set;

theorem
  for C being with_all_isomorphisms lattice-wise category for a,b
  being Object of C for f being Morphism of a,b st @f is isomorphic holds f is
  iso
proof
  let C be with_all_isomorphisms lattice-wise category;
  let a,b be Object of C;
  let f be Morphism of a,b;
  assume
A1: @f is isomorphic;
  then consider g being monotone Function of latt b, latt a such that
A2: @f*g = id latt b and
A3: g*@f = id latt a by YELLOW16:15;
A4: @f in <^a,b^> by A1,Def8;
A5: g is isomorphic by A2,A3,YELLOW16:15;
  then
A6: g in <^b,a^> by Def8;
  reconsider g as Morphism of b,a by A5,Def8;
A7: @g = g by A6,Def7;
  idm b = id latt b by Th2;
  then
 f*g = idm b by A2,A4,A6,A7,Th3;
  then
A8: g is_right_inverse_of f;
  idm a = id latt a by Th2;
  then
 g*f = idm a by A3,A4,A6,A7,Th3;
  then
A9: g is_left_inverse_of f;
  then f is retraction coretraction by A8;
  hence f*f" = idm b & f"*f = idm a by A4,A6,A9,A8,ALTCAT_3:def 4;
end;
