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 Th77:
  for t, u, v, w st t LD-= u & v LD-= w holds t '&' v LD-= u '&' w
proof
  deffunc F( GRZ-formula, GRZ-formula ) = $1 '&' $2;
  defpred P[ GRZ-formula, GRZ-formula ] means $1 '=' $2 is LD-provable;
  A2: for t, u, v st P[t, u] & P[u, v] holds P[t, v] by Th74;
  A3: for t, u holds P[F(t,u), F(u,t)];
  A4: for t, u, v st P[t, u] holds P[F(t,v), F(u,v)] by Lm77a;
  for t, u, v, w st P[t, u] & P[v, w] holds P[F(t, v), F(u, w)]
      from BinReplace(A2, A3, A4);
  hence thesis;
end;
