
theorem Th3:
  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 & A is
  coretraction holds (A")" = A
proof
  let C be category, o1,o2 be Object of C such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume
A3: A is retraction & A is coretraction;
  then A" is_left_inverse_of A by A1,A2,Def4;
  then
A4: A" is retraction;
A5: A" is_right_inverse_of A by A1,A2,A3,Def4;
  then A" is coretraction;
  then
A6: (A")" is_right_inverse_of A" by A1,A2,A4,Def4;
  thus (A")" = idm o2 * ((A")") by A1,ALTCAT_1:20
    .= A * A" * (A")" by A5
    .= A * (A" * (A")") by A1,A2,ALTCAT_1:21
    .= A * idm o1 by A6
    .= A by A1,ALTCAT_1:def 17;
end;
