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
  for x, y st x is LD-provable holds x 'or' y is LD-provable
proof
  let x, y;
  assume A1: x is LD-provable;
  consider u such that A2: y = LD-EqClassOf u by Th88;
  LD-EqClassOf ('not' (u '&' 'not' u))
      = 'not' LD-EqClassOf (u '&' 'not' u) by Def91
      .= 'not' ((LD-EqClassOf u) '&' (LD-EqClassOf 'not' u)) by Def92
      .= 'not' (y '&' 'not' y) by A2, Def91;
  then A5: x '&' ('not' y 'or' y) is LD-provable by A1, Th91;
  x '&' ('not' y 'or' y) = (x '&' 'not' y) 'or' (x '&' y) by Th104
      .= ((x '&' 'not' y) 'or' x) '&' ((x '&' 'not' y) 'or' y) by Th105;
  then A6: (x '&' 'not' y) 'or' y is LD-provable by A5, Th91;
  (x '&' 'not' y) 'or' y = (y 'or' x) '&' (y 'or' 'not' y) by Th105;
  hence thesis by A6, Th91;
end;
