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;
