
theorem Th18:
  for C being category, o1,o2,o3 being Object of C st <^o1,o2^> <>
  {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} for A being Morphism of o1,o2, B being
Morphism of o2,o3 st A is retraction & B is retraction holds B*A is retraction
proof
  let C be category, o1,o2,o3 be Object of C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {} and
A3: <^o3,o1^> <> {};
A4: <^o2,o1^> <> {} by A2,A3,ALTCAT_1:def 2;
A5: <^o3,o2^> <> {} by A1,A3,ALTCAT_1:def 2;
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A6: A is retraction and
A7: B is retraction;
  consider A1 being Morphism of o2,o1 such that
A8: A1 is_right_inverse_of A by A6;
  consider B1 being Morphism of o3,o2 such that
A9: B1 is_right_inverse_of B by A7;
  consider G being Morphism of o3,o1 such that
A10: G = A1 * B1;
  take G;
  (B * A) * G = B * (A * (A1 * B1)) by A1,A2,A3,A10,ALTCAT_1:21
    .= B * ((A * A1) * B1) by A1,A4,A5,ALTCAT_1:21
    .= B * (idm o2 * B1) by A8
    .= B * B1 by A5,ALTCAT_1:20
    .= idm o3 by A9;
  hence thesis;
end;
