
theorem
  for A being trivial set holds
    IdPrefSpace A = PrefSpace A
  proof
    let A be trivial set;
    per cases;
    suppose
A0:   A is non empty;
A1:   the carrier of IdPrefSpace A = the carrier of PrefSpace A by Def5;
A2:   the ToleranceRel of IdPrefSpace A = id the carrier of PrefSpace A
        by Def5
        .= the ToleranceRel of PrefSpace A by Lemma1C,A0;
      the InternalRel of IdPrefSpace A = the InternalRel of PrefSpace A
        by Def5;
      hence thesis by A1,A2,Def5;
    end;
    suppose A is empty;
      hence thesis by ZZ;
    end;
  end;
