
theorem Th22:
  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 & f is retraction coretraction holds f9" = f"
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;
  let f be Morphism of a,b, f9 be Morphism of b9,a9 such that
A8: f9 = f and
A9: f is retraction coretraction;
  reconsider g = f" as Morphism of a9, b9 by A1,A4,A5,Th7;
A10: f"*f = idm a by A2,A3,A9,ALTCAT_3:2;
A11: f*f" = idm b by A2,A3,A9,ALTCAT_3:2;
A12: f9*g = idm a by A1,A2,A3,A4,A5,A8,A10,Th9;
A13: g*f9 = idm b by A1,A2,A3,A4,A5,A8,A11,Th9;
A14: f9*g = idm a9 by A1,A4,A12,Th10;
A15: g*f9 = idm b9 by A1,A5,A13,Th10;
A16: f9 is retraction coretraction by A1,A2,A3,A4,A5,A8,A9,Lm1;
A17: g is_left_inverse_of f9 by A15;
  g is_right_inverse_of f9 by A14;
  hence thesis by A2,A3,A6,A7,A16,A17,ALTCAT_3:def 4;
end;
