
theorem Th31:
  for A being non empty set, a being Object of EnsCat A holds idm a = id a
proof
  let A be non empty set, a be Object of EnsCat A;
  <^a,a^> = Funcs(a, a) by ALTCAT_1:def 14;
  then reconsider e = id a as Morphism of a,a by FUNCT_2:126;
  now
    let b being Object of EnsCat A such that
A1: <^a,b^> <> {};
    let f be Morphism of a,b;
A2: <^a,b^> = Funcs(a, b) by ALTCAT_1:def 14;
    then reconsider g = f as Function;
A3: dom g = a by A1,A2,Th30;
    thus f*e = g* id a by A1,ALTCAT_1:def 12
      .= f by A3,RELAT_1:52;
  end;
  hence thesis by ALTCAT_1:def 17;
end;
