
theorem
  for C being category, o1, o2 being Object of C, A being Morphism of o1
, o2 st A is coretraction & A is epi & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
  A is iso
proof
  let C be category, o1, o2 be Object of C, A be Morphism of o1,o2;
  assume that
A1: A is coretraction and
A2: A is epi and
A3: <^o1,o2^> <> {} and
A4: <^o2,o1^> <> {};
  consider B being Morphism of o2,o1 such that
A5: B is_left_inverse_of A by A1;
  A * (B * A) = A * (idm o1) by A5;
  then A * (B * A) = A by A3,ALTCAT_1:def 17;
  then A * (B * A) = idm o2 * A by A3,ALTCAT_1:20;
  then
A6: <^o2,o2^> <> {} & (A * B) * A = idm o2 * A by A3,A4,ALTCAT_1:19,21;
  then A * B = idm o2 by A2;
  then
A7: B is_right_inverse_of A;
  then
A8: A is retraction;
  then
A9: A"*A = B * A by A1,A3,A4,A5,A7,Def4
    .= idm o1 by A5;
  A*A" = A * B by A1,A3,A4,A5,A7,A8,Def4
    .= idm o2 by A2,A6;
  hence thesis by A9;
end;
