theorem Th33:
  for a being Element of C, the bool-sort of S
  for x being boolean object holds
  \nota = 'not' x iff a = x
  proof
    let a be Element of C, the bool-sort of S;
    a in (the Sorts of C).the bool-sort of S;
    then a in BOOLEAN by AOFA_A00:def 32;
    then reconsider b = a as boolean object;
    let x be boolean object;
    hereby
      assume \nota = 'not' x;
      then 'not' b = 'not' x by AOFA_A00:def 32;
      hence a = x;
    end;
    assume a = x;
    hence \nota = 'not' x by AOFA_A00:def 32;
  end;
