theorem
  for A, R, t st A, R |- t holds t in A or ex S st [S, t] in R & A, R |- S
proof
  let A, R, t;
  assume that
    A1: A, R |- t and
    A2: not t in A;
  consider P such that
    A3: t in rng P and
    A4: P is (A, R)-correct by A1;
  consider a such that
    A5: a in dom P and
    A6: P.a = t by A3, FUNCT_1:def 3;
  reconsider n = a as Element of NAT by A5;
  P, n is_a_correct_step_wrt A, R by A4, A5;
  then consider Q being GRZ-formula-finset such that
    A10: [Q, P.n] in R and
    A11: for q st q in Q ex k
            st k in dom P & k < n & P.k = q by A2, A6;
  take Q;
  thus [Q, t] in R by A6, A10;
  let u;
  assume u in Q;
  then consider k such that
    A15: k in dom P and
         k < n and
    A17: P.k = u by A11;
  u in rng P by A15, A17, FUNCT_1:3;
  hence A, R |- u by A4;
end;
