reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;
reserve B,P for non empty Subset of X;

theorem
  X is BCK-algebra of 1,0,0,0 iff X is BCK-algebra of 0,0,0,0 & X is
  BCK-algebra of 0,1,0,1
proof
  thus X is BCK-algebra of 1,0,0,0 implies X is BCK-algebra of 0,0,0,0 & X is
  BCK-algebra of 0,1,0,1
  proof
    assume X is BCK-algebra of 1,0,0,0;
    then X is BCK-implicative BCK-algebra by Th50;
    hence thesis by Th38,Th49;
  end;
  assume X is BCK-algebra of 0,0,0,0 & X is BCK-algebra of 0,1,0,1;
  then
  X is commutative BCK-algebra & X is BCK-positive-implicative BCK-algebra
  by Th38,Th49;
  then X is BCK-implicative BCK-algebra by BCIALG_3:34;
  hence thesis by Th50;
end;
