
theorem Th23:
  for A, B being category st A, B are_opposite
  for a, b being Object of A st <^a,b^> <> {} & <^b,a^> <> {}
  for a9, b9 being Object of B st a9 = a & b9 = b
  for f being Morphism of a,b, f9 being Morphism of b9,a9 st f9 = f
  holds f is iso iff f9 is iso
proof
  let A, B be category such that
A1: A, B are_opposite;
  let a, b be Object of A such that
A2: <^a,b^> <> {} and
A3: <^b,a^> <> {};
  let a9, b9 be Object of B such that
A4: a9 = a and
A5: b9 = b;
A6: <^b9,a9^> = <^a,b^> by A1,A4,A5,Th9;
A7: <^a9,b9^> = <^b,a^> by A1,A4,A5,Th9;
  now
    let A, B be category such that
A8: A, B are_opposite;
    let a, b be Object of A such that
A9: <^a,b^> <> {} and
A10: <^b,a^> <> {};
    let a9, b9 be Object of B such that
A11: a9 = a and
A12: b9 = b;
    let f be Morphism of a,b, f9 be Morphism of b9,a9 such that
A13: f9 = f;
    assume
A14: f is iso;
    then
A15: f*f" = idm b;
A16: f"*f = idm a by A14;
    f is retraction coretraction by A14,ALTCAT_3:5;
    then
A17: f9" = f" by A8,A9,A10,A11,A12,A13,Th22;
A18: idm a = idm a9 by A8,A11,Th10;
A19: idm b = idm b9 by A8,A12,Th10;
A20: f9*f9" = idm a9 by A8,A9,A10,A11,A12,A13,A16,A17,A18,Th9;
    f9"*f9 = idm b9 by A8,A9,A10,A11,A12,A13,A15,A17,A19,Th9;
    hence f9 is iso by A20;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,A7;
end;
