reserve C for category,
  o1, o2, o3 for Object of C;

theorem Th32:
  for A, B being category, F being contravariant Functor of A, B
for o1, o2 being Object of A, a being Morphism of o1, o2 st F is full faithful
  & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F.a is iso holds a is iso
proof
  let A, B be category, F be contravariant Functor of A, B, o1, o2 be Object
  of A, a be Morphism of o1, o2 such that
A1: F is full faithful and
A2: <^o1,o2^> <> {} & <^o2,o1^> <> {} and
A3: F.a is iso;
  <^F.o1,F.o2^> <> {} & <^F.o2,F.o1^> <> {} by A2,FUNCTOR0:def 19;
  then F.a is retraction coretraction by A3,ALTCAT_3:6;
  then a is retraction coretraction by A1,A2,Th30,Th31;
  hence thesis by A2,ALTCAT_3:6;
end;
