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 Th57:
  for Fml for B being Subset of Fml, R being Rule of Fml, a st B, R |- a
    holds a in Fml
proof
  let Fml;
  let B be Subset of Fml;
  let R be Rule of Fml;
  let a;
  assume B, R |- a;
  then consider P such that A1: a in rng P and A2: P is (B,R)-correct;
  P is Formula-sequence of Fml by A2, Lm55;
  then P is FinSequence of Fml by Def5;
  then rng P c= Fml by FINSEQ_1:def 4;
  hence thesis by A1;
end;
