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 Th91:
  for x, y holds x '&' y is LD-provable iff x is LD-provable & y is LD-provable
proof
  let x, y;
  consider t, u such that
    A2: x = LD-EqClassOf t & y = LD-EqClassOf u and
    A3: x '&' y = LD-EqClassOf t '&' u by Def92;
  thus x '&' y is LD-provable implies x is LD-provable & y is LD-provable
    proof
    assume x '&' y is LD-provable;
    then t '&' u is LD-provable by A3, Th90;
    then t is LD-provable & u is LD-provable by Th60;
    hence thesis by A2;
    end;
  assume x is LD-provable & y is LD-provable;
  then t is LD-provable & u is LD-provable by A2, Th90;
  then t '&' u is LD-provable by Th60;
  hence x '&' y is LD-provable by A3;
end;
