
theorem Th10:
  for A,B being category st A, B are_opposite
  for a being Object of A, b being Object of B st a = b holds idm a = idm b
proof
  let A,B be category such that
A1: A, B are_opposite;
  let a be Object of A, b be Object of B such that
A2: a = b;
  reconsider i = idm b as Morphism of a,a by A1,A2,Th9;
  now
    let c be Object of A;
    assume
A3: <^a,c^> <> {};
    let f be Morphism of a,c;
    reconsider d = c as Object of B by A1;
A4: <^a,c^> = <^d,b^> by A1,A2,Th9;
    reconsider g = f as Morphism of d,b by A1,A2,Th9;
    thus f*i = (idm b)*g by A1,A2,A3,Th9
      .= f by A3,A4,ALTCAT_1:20;
  end;
  hence thesis by ALTCAT_1:def 17;
end;
