
theorem Th15:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
  & <^o2,o1^> <> {} for A being Morphism of o1,o2 st A is retraction holds A is
  epi
proof
  let C be category, o1,o2 be Object of C;
  assume
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume A is retraction;
  then consider R being Morphism of o2,o1 such that
A2: R is_right_inverse_of A;
  let o be Object of C;
  assume
A3: <^o2,o^> <> {};
  let B,C be Morphism of o2,o;
  assume
A4: B * A = C * A;
  thus B = B * idm o2 by A3,ALTCAT_1:def 17
    .= B * (A * R) by A2
    .= C * A * R by A1,A3,A4,ALTCAT_1:21
    .= C * (A * R) by A1,A3,ALTCAT_1:21
    .= C * idm o2 by A2
    .= C by A3,ALTCAT_1:def 17;
end;
