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 Th101:
  for x, y, z st x => y is LD-provable & y => z is LD-provable holds
      x => z is LD-provable
proof
  let x, y, z;
  assume that
    A1: x => y is LD-provable and
    A2: y => z is LD-provable;
  x = x '&' y by A1, Th92
      .= x '&' (y '&' z) by A2, Th92
      .= (x '&' y) '&' z by Th98
      .= x '&' z by A1, Th92;
  hence thesis by Th92;
end;
