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 Th53:
  for B, R, B1, a st B1, R |- a & for b st b in B1 holds B, R |- b
    holds B, R |- a
proof
  let B, R, B1, a;
  assume that
    A1: B1, R |- a and
    A2: for b st b in B1 holds B, R |- b;
  consider S such that A3: a in S and A4: S is (B1, R)-derivable by A1, Th45;
  set S1 = B1 /\ S;
  for b st b in S1 holds b in B1 by XBOOLE_0:def 4;
  then for b st b in S1 holds B, R |- b by A2;
  then consider S2 such that
    A7: S1 c= S2 and
    A8: S2 is (B, R)-derivable by Th47;
  S is (S2, R)-derivable by A4, A7, Th48;
  then A10: S2 \/ S is (B, R)-derivable by A8, Th52;
  a in S2 \/ S by A3, XBOOLE_0:def 3;
  hence thesis by A10;
end;
