 By contraposing  Proposition 3.5  it thus follows that in order to show that a set \(B\) is non-computable (or non-c.e.) it suffices to show that a known non-computable (or non-c.e.) set can be reduced to it. As an initial example, observe that the Diagonal Halting Problem \(K = \{i : \phi_i(i) \darrow\}\) is reducible to the Halting Problem \(\HP = \{\langle i,n \rangle : \phi_i(n) \darrow\}\) by the reduction \(f(x) = \langle x,x \rangle\)—i.e., the computable pairing function of \(x\) with itself is a many-one reduction showing \(K \leq_m \HP\). Thus since \(\HP\) is known to be non-computable, this gives another proof that \(K\) also is not computable.