Disprove unrealistic speed-up of total Turing machines
Suppose that the input alphabet is $\{0,1\}$, and consider the language $L_1 = 0^*$. We can easily construct a Turing machine $T_1$ such that $S_{T_1}(I) = |I|+1$. On the other hand, $S_{T_2}(I_c) \geq |I_c|+1$. Since $|I_c|=|I_1|+1+|I_2|$, we get
$$
S_{T_2}(I_c) \geq |I_1|+1+|I_2|+1 = S_{T_1}(I_1) + S_{T_1}(I_2).
$$