theorem
  for B, t, a holds a is (B \/ {t})-axiomatic iff a is B-axiomatic or a = t
proof
  let B, t, a;
  set C = B \/ {t};
  thus a is C-axiomatic implies a is B-axiomatic or a = t
  proof
    assume a is C-axiomatic;
    then a in B or a in { t } by XBOOLE_0:def 3;
    hence thesis by TARSKI:def 1;
  end;
  assume a is B-axiomatic or a = t;
  then a in B or a in { t } by TARSKI:def 1;
  hence thesis by XBOOLE_0:def 3;
end;
