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 t, u, v st t => u is LD-provable & u => v is LD-provable holds
      t => v is LD-provable
proof
  let t, u, v;
  assume A1: t => u is LD-provable & u => v is LD-provable;
  set x = LD-EqClassOf t;
  set y = LD-EqClassOf u;
  set z = LD-EqClassOf v;
  A2: LD-EqClassOf (t => u) = x => y & LD-EqClassOf (u => v) = y => z by Th97;
  LD-EqClassOf (t => v) = x => z by Th97;
  hence t => v is LD-provable by A1, A2, Th90, Th101;
end;
