
theorem Th20:
  for C being category, o1, o2 being Object of C, A being Morphism
  of o1,o2 st A is retraction & A is mono & <^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 retraction and
A2: A is mono and
A3: <^o1,o2^> <> {} and
A4: <^o2,o1^> <> {};
  consider B being Morphism of o2,o1 such that
A5: B is_right_inverse_of A by A1;
  A * B * A = (idm o2) * A by A5;
  then A * (B * A) = (idm o2) * A by A3,A4,ALTCAT_1:21;
  then A * (B * A) = A by A3,ALTCAT_1:20;
  then
A6: <^o1,o1^> <> {} & A * (B * A) = A * idm o1 by A3,ALTCAT_1:19,def 17;
  then B * A = idm o1 by A2;
  then
A7: B is_left_inverse_of A;
  then
A8: A is coretraction;
  then
A9: A*A" = A * B by A1,A3,A4,A5,A7,Def4
    .= idm o2 by A5;
  A"*A = B * A by A1,A3,A4,A5,A7,A8,Def4
    .= idm o1 by A2,A6;
  hence thesis by A9;
end;
