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

theorem Th1:
  for v being Morphism of o1, o2, u being Morphism of o1, o3 for f
being Morphism of o2, o3 st u = f * v & f" * f = idm o2 & <^o1,o2^> <> {} & <^
  o2,o3^> <> {} & <^o3,o2^> <> {} holds v = f" * u
proof
  let v be Morphism of o1, o2, u be Morphism of o1, o3, f be Morphism of o2,
  o3 such that
A1: u = f * v and
A2: f" * f = idm o2 and
A3: <^o1,o2^> <> {} and
A4: <^o2,o3^> <> {} & <^o3,o2^> <> {};
  thus f" * u = f" * f * v by A1,A3,A4,ALTCAT_1:21
    .= v by A2,A3,ALTCAT_1:20;
end;
