
theorem Th37:
  for C being para-functional semi-functional category
  for a being Object of C st id the_carrier_of a in <^a,a^>
  holds idm a = id the_carrier_of a
proof
  let C be para-functional semi-functional category;
  let a be Object of C;
  assume id the_carrier_of a in <^a,a^>;
  then reconsider f = id the_carrier_of a as Morphism of a,a;
  now
    let b be Object of C such that
A1: <^a,b^> <> {};
    let g being Morphism of a,b;
A2: dom g = the_carrier_of a by A1,Th35;
    thus g*f = g* id the_carrier_of a by A1,Th36
      .= g by A2,RELAT_1:52;
  end;
  hence thesis by ALTCAT_1:def 17;
end;
