reserve k,m,n for Element of NAT,
  i, j for Nat,
  a, b, c for object,
  X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s for FinSequence;
reserve t, u, v, w for GRZ-formula;
reserve R, R1, R2 for GRZ-rule;
reserve A, A1, A2 for non empty Subset of GRZ-formula-set;
reserve B, B1, B2 for Subset of GRZ-formula-set;
reserve P, P1, P2 for GRZ-formula-sequence;
reserve S, S1, S2 for GRZ-formula-finset;

theorem Th61:
  for A, t, u st A, GRZ-rules |- t & A, GRZ-rules |- t '=' u holds
      A, GRZ-rules |- u
proof
  let A, t, u;
  assume A1: A, GRZ-rules |- t & A, GRZ-rules |- t '=' u;
  set S = {t, t '=' u};
  for a st a in S holds a in GRZ-formula-set
    proof
    let a;
    assume a in S;
    then a = t or a = t '=' u by TARSKI:def 2;
    hence thesis;
    end;
  then S c= GRZ-formula-set;
  then reconsider S as GRZ-formula-finset;
  A3: A, GRZ-rules |- S by A1, TARSKI:def 2;
  [S, u] in GRZ-MP;
  then [S, u] in GRZ-rules by Def35;
  hence A, GRZ-rules |- u by A3, Th48;
end;
