reserve x, y for set;

theorem
  for C being lattice-wise category for a,b being Object of C st <^a,b^>
  <> {} & <^b,a^> <> {} for f being Morphism of a,b st f is iso holds @f is
  isomorphic
proof
  let C be lattice-wise category;
  let a,b be Object of C;
  assume
A1: <^a,b^> <> {} & <^b,a^> <> {};
  let f be Morphism of a,b such that
A2: f*f" = idm b & f"*f = idm a;
A3: idm a = id latt a & idm b = id latt b by Th2;
  @f*@(f") = f*f" & @(f")*@f = f"*f by A1,Th3;
  hence thesis by A2,A3,YELLOW16:15;
end;
