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;
reserve x, y, z for LD-EqClass;

theorem Th98:
  for x, y, z holds (x '&' y) '&' z = x '&' (y '&' z)
proof
  let x, y, z;
  consider t, u such that
    A1: x = LD-EqClassOf t and
    A2: y = LD-EqClassOf u and
    A3: x '&' y = LD-EqClassOf (t '&' u) by Def92;
  consider v such that
    A5: z = LD-EqClassOf v by Th88;
  A10: (t '&' (u '&' v)) '=' ((t '&' u) '&' v) is LD-provable;
  thus (x '&' y) '&' z = LD-EqClassOf ((t '&' u) '&' v) by A3, A5, Def92
      .= LD-EqClassOf (t '&' (u '&' v)) by A10, Th80, Def76
      .= (LD-EqClassOf t) '&' (LD-EqClassOf (u '&' v)) by Def92
      .= x '&' (y '&' z) by A1, A2, A5, Def92;
end;
