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 Th45:
  for B, R, t holds B, R |- t iff ex S st t in S & S is (B, R)-derivable
proof
  let B, R, t;
  thus B, R |- t implies ex S st (t in S & S is (B, R)-derivable)
  proof
    assume B, R |- t;
    then consider P such that A1: t in rng P and A2: P is (B, R)-correct;
    take S = rng P;
    thus t in S by A1;
    thus S is (B, R)-derivable by A2;
  end;
  given S such that A10: t in S and A11: S is (B, R)-derivable;
  consider P such that A12: S = rng P and A13: P is (B, R)-correct by A11;
  thus thesis by A10, A12, A13;
end;
