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;

theorem Th74:
  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 that
    A1: t '=' u is LD-provable and
    A2: u '=' v is LD-provable;
  A3: u '=' t is LD-provable by A1, Th70, XBOOLE_1:7;
  set w = (u '=' v) '=' (t '=' v);
  (u '=' t) => w is LD-provable;
  then w is LD-provable by A3, Th62;
  hence thesis by A2, Th61;
end;
