reserve P for non empty Poset,
  i, j, k for Element of P;
reserve S for non void non empty ManySortedSign;
reserve OAF for OrderedAlgFam of P, S;
reserve B for Binding of OAF;

theorem Th1:
  i >= j & j >= k implies bind (B,j,k) ** bind (B,i,j) = bind (B,i, k)
proof
  assume
A1: i >= j & j >= k;
  then
A2: ex f1 be ManySortedFunction of OAF.i, OAF.j, f2 be ManySortedFunction of
  OAF.j, OAF.k st f1 = B.(j,i) & f2 = B.(k,j) & B.(k,i) = f2 ** f1 & f1
  is_homomorphism OAF.i, OAF.j by Def2;
  bind (B,j,k) = B.(k,j) & bind (B,i,j) = B.(j,i) by A1,Def3;
  hence thesis by A1,A2,Def3,ORDERS_2:3;
end;
