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

theorem Th2:
  for v being Morphism of o2, o3, u being Morphism of o1, o3 for f
being Morphism of o1, o2 st u = v * f & f * f" = idm o2 & <^o1,o2^> <> {} & <^
  o2,o1^> <> {} & <^o2,o3^> <> {} holds v = u * f"
proof
  let v be Morphism of o2, o3, u be Morphism of o1, o3, f be Morphism of o1,
  o2 such that
A1: u = v * f and
A2: f * f" = idm o2 and
A3: <^o1,o2^> <> {} & <^o2,o1^> <> {} and
A4: <^o2,o3^> <> {};
  thus u * f" = v * (f * f") by A1,A3,A4,ALTCAT_1:21
    .= v by A2,A4,ALTCAT_1:def 17;
end;
