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 Th104:
  for x, y, z holds x '&' (y 'or' z) = ((x '&' y) 'or' (x '&' z))
proof
  let x, y, z;
  consider t such that A1: x = LD-EqClassOf t by Th88;
  consider u such that A2: y = LD-EqClassOf u by Th88;
  consider v such that A3: z = LD-EqClassOf v by Th88;
  (t '&' (u 'or' v)) '=' ((t '&' u) 'or' (t '&' v)) is LD-provable;
  then A5: LD-EqClassOf (t '&' (u 'or' v))
      = LD-EqClassOf ((t '&' u) 'or' (t '&' v)) by Th80, Def76;
  thus x '&' (y 'or' z)
      = (LD-EqClassOf t) '&' (LD-EqClassOf (u 'or' v)) by A1, A2, A3, Th96
      .= LD-EqClassOf((t '&' u) 'or' (t '&' v)) by A5, Def92
      .= (LD-EqClassOf (t '&' u)) 'or' (LD-EqClassOf (t '&' v)) by Th96
      .= (LD-EqClassOf (t '&' u))
            'or' ((LD-EqClassOf t) '&' (LD-EqClassOf v)) by Def92
      .= ((x '&' y) 'or' (x '&' z)) by A1, A2, A3, Def92;
end;
