Symbol Reduction of Turing Machine

Theorem
Let $T$ be a Turing machine with input symbols $\Sigma$ and blank symbol $B$.

Then there exists a Turing machine $T'$ such that:
 * The input symbols of $T'$ are $\Sigma$
 * The tape symbols of $T'$ are $\Sigma \cup \set B$
 * The language accepted by $T'$ is exactly the language accepted by $T$
 * The inputs on which $T'$ halts are exactly the inputs on which $T$ halts