$$
\begin{array}{l} \left\| \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right\| _ {F} ^ {2} \\ = \left\| \left[ \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) + \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right. \right. \\ \left. - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) + \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right] \\ - \left[ \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) + \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right. \\ \left. \left. - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) + \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right] \right\| _ {F} ^ {2} \\ = \left\| \left[ \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right] \right. \\ + \left[ \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right] \\ + \left[ \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right] \tag {56} \\ + \left[ \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right] \| _ {F} ^ {2} \\ \leq 4 \left\| \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ + 4 \left\| \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right\| _ {F} ^ {2} \\ + 4 \left\| \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right\| _ {F} ^ {2} \\ + 4 \left\| \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \right\| _ {F} ^ {2} \\ \leq 4 \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} + 4 \left(L _ {Z _ {p ^ {\prime}}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1} \right\| _ {F} ^ {2} + 8 \zeta^ {2} \\ \end{array}
$$