
theorem Th16:
  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 coretraction holds A is
  mono
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 coretraction;
  then consider R being Morphism of o2,o1 such that
A2: R is_left_inverse_of A;
  let o be Object of C;
  assume
A3: <^o,o1^> <> {};
  let B,C be Morphism of o,o1;
  assume
A4: A * B = A * C;
  thus B = idm o1 * B by A3,ALTCAT_1:20
    .= R * A * B by A2
    .= R * (A * C) by A1,A3,A4,ALTCAT_1:21
    .= R * A * C by A1,A3,ALTCAT_1:21
    .= idm o1 * C by A2
    .= C by A3,ALTCAT_1:20;
end;
