The theorems we have which we will show is Theorem : There exists a Turing Machine $\mathcal{U}$ such that for every $x,\alpha\in \{0,1\}^*$ $\mathcal{U}(x,\alpha)=M_{\alpha}(x)$ where $M_{\alpha}$ denotes the turing machine represented by $\alpha$. Moreover, if $M_{\alpha}$ halts on input $x$ within $T$ steps then $\mathcal{U}(x,\alpha)$ halts within $cT\log T$ steps where $C$ is a number of independent of $|x|$ and depending only on $M_{\alpha}'s$ alphabet size, number of tapes and number of states For this, we need some equivalency between some special Turing machine models Theorem 1: For every $f:\{0,1\}^*\to \{0,1\}$ and time-constructible $T:\mathbb{N}\to \mathbb{N}$ if $f$ is computable in time $T(n)$ by a Turing Machine $M$ using alphabet $\Gamma$ then it is computable in time $4\log|\Gamma|T(n)$ by a Turing Machine $\hat{M}$ using the alphabet $\{\triangleright, \square,0,1\}$ Theorem 2: Define a single-tape Turing machine to be a Turing Machine th...