theorem
  for a being boolean object
  for t being Element of T, the bool-sort of S holds
  \nott value_at(C,u) = 'not' a iff t value_at(C,u) = a
  proof
    let a be boolean object;
    let t be Element of T, the bool-sort of S;
    hereby
      assume \nott value_at(C,u) = 'not' a;
      then \not(t value_at(C,u)) = 'not' a by Th47;
      then 'not' (t value_at(C,u)) = 'not' a by AOFA_A00:def 32;
      hence t value_at(C,u) = a;
    end;
    assume A1: t value_at(C,u) = a;
    \not(t value_at(C,u)) = \nott value_at(C,u) by Th47;
    hence \nott value_at(C,u) = 'not' a by A1,AOFA_A00:def 32;
  end;
