 Proof: Suppose we have a Turing Complete system F. By Definition any computation \(T_i(x)\) on a Turing machine can be emulated in F and vice versa. There will be a special universal Turing machine \(U_F\) that emulates the computation \(T_i(x)\) in F: \(U_F(\overline{T}_i^Fx)\). In principle \(\overline{T}_i^F\) might use a very inefficient way to code programs such that \(\overline{T}_i^F\) can have any length. Observe that the code for any other universal machine \(U_j\) emulated by \(U_F\) has constant length, i.e., \(l(\overline{U}_j^F)=c\). Since \(U_F\) is universal we can also compute: 