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 holds (x => y is LD-provable & y => x is LD-provable) iff x = y
proof
  let x, y;
  thus (x => y is LD-provable & y => x is LD-provable) implies x = y
  proof
    assume that A1: x => y is LD-provable and A2: y => x is LD-provable;
    thus x = x '&' y by A1, Th92 .= y by A2, Th92;
  end;
  assume x = y;
  hence x => y is LD-provable & y => x is LD-provable by Th92;
end;
