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

theorem Th18:
  for A, B being transitive with_units non empty AltCatStr for F
being covariant Functor of A, B for o1, o2 being Object of A, a being Morphism
  of o1, o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds F.a is
  retraction
proof
  let A, B be transitive with_units non empty AltCatStr, F be covariant
  Functor of A, B, o1, o2 be Object of A, a be Morphism of o1, o2 such that
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  assume a is retraction;
  then consider b being Morphism of o2, o1 such that
A2: b is_right_inverse_of a;
  take F.b;
  a * b = idm o2 by A2;
  hence (F.a) * (F.b) = F.idm o2 by A1,FUNCTOR0:def 23
    .= idm F.o2 by FUNCTOR2:1;
end;
