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;
 reserve C for Extension of B;
 reserve E for Extension of R;

theorem Th48:
  for B, B1, R, S st S is (B, R)-derivable & B /\ S c= B1
      holds S is (B1, R)-derivable
proof
  let B, B1, R, S;
  assume that A1: S is (B, R)-derivable and A2: B /\ S c= B1;
  consider P such that A3: S = rng P and A4: P is (B,R)-correct by A1;
  P is (B1, R)-correct
  proof
    let k;
    assume A6: k in dom P;
    then A7: P.k in S by A3, FUNCT_1:3;
    P, k is_a_correct_step_wrt B, R by A4, A6;
    then per cases;
      suppose P.k in B;
      then P.k in B /\ S by A7, XBOOLE_0:def 4;
      hence thesis by A2;
      end;
      suppose ex Q being Formula-finset st
        ([Q, P.k] in R
            & for t st t in Q ex m st m in dom P & m < k & P.m = t);
      then consider Q being Formula-finset such that
        A10: [Q, P.k] in R and
        A11: for t st t in Q ex m st m in dom P & m < k & P.m = t;
      thus thesis by A10, A11;
    end;
  end;
  hence thesis by A3;
end;
