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 Th60:
  for A, t, u holds
    A, GRZ-rules |- t '&' u iff A, GRZ-rules |- t & A, GRZ-rules |- u
proof
  let A, t, u;
  thus A, GRZ-rules |- t '&' u implies A, GRZ-rules |- t & A, GRZ-rules |- u
    proof
    assume A1: A, GRZ-rules |- t '&' u;
    set S = {t '&' u};
    for a st a in S holds a in GRZ-formula-set
      proof
      let a;
      assume a in S;
      then a = t '&' u by TARSKI:def 1;
      hence thesis;
      end;
    then S c= GRZ-formula-set;
    then reconsider S as GRZ-formula-finset;
    A2: A, GRZ-rules |- S by A1, TARSKI:def 1;
    [S, t] in GRZ-ConjElimL;
    then [S, t] in GRZ-rules by Def35;
    hence A, GRZ-rules |- t by A2, Th48;
    [S, u] in GRZ-ConjElimR;
    then [S, u] in GRZ-rules by Def35;
    hence A, GRZ-rules |- u by A2, Th48;
    end;
  assume that A10: A, GRZ-rules |- t and A11: A, GRZ-rules |- u;
  set S1 = {t, u};
  for a st a in S1 holds a in GRZ-formula-set
    proof
    let a;
    assume a in S1;
    then a = t or a = u by TARSKI:def 2;
    hence thesis;
    end;
  then S1 c= GRZ-formula-set;
  then reconsider S1 as GRZ-formula-finset;
  A12: A, GRZ-rules |- S1 by A10, A11, TARSKI:def 2;
  [S1, t '&' u] in GRZ-ConjIntro;
  then [S1, t '&' u] in GRZ-rules by Def35;
  hence A, GRZ-rules |- t '&' u by A12, Th48;
end;
