Proof:

Let $F:\: [0,1] \times [0,1] \rightarrow X$ be a homotopy between $\gamma_1^\prime$ and $\gamma_1^{\prime \prime}$ so that $F(0,t)=\gamma_0(1),\\ F(1,t)=\gamma_1^{\prime}(1) =\gamma_1^{\prime \prime}(1)$. The homotopy we seek is the map $H(s, t)$ given by

$$H(s, t) = \left\{\begin{array}{lll} \gamma_0(2t), & & 0 \leq t \leq 1/2 \\ F(s, 2t - 1), & & 1/2 \leq t \leq 1 \\ \end{array} \right.$$

It can be checked that the definition is meaningful along $[0,1] \times \{ \frac{1}{2} \}$ and the continuity of $H$ follows by the gluing lemma. The reader may complete the proof by verifying that

$$H(0,t)=\gamma_0 \ast \gamma_1^\prime;\quad H(1,t)=\gamma_0 \ast \gamma_1^{\prime \prime}.\eqno\square$$