reserve i, j, k, l, m, n for Nat,
  a, b, c, t, u for object,
  X, Y, Z for set,
  D, D1, D2, Fml for non empty set;
reserve p, q, r, s for FinSequence;
 reserve R, R1, R2 for Rule;
 reserve A, A1, A2 for non empty set;
 reserve B, B1, B2 for set;
 reserve P, P1, P2 for Formula-sequence;
 reserve S, S1, S2 for Formula-finset;

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;
