Let $(X_t)$ a stochastic process, and denote $$p(t_1,x_1,...,t_n,x_n)=\frac{\partial ^n}{\partial x_1...\partial x_n}\mathbb P\{X_{t_1}\leq x_1,...,X_{t_n}\leq x_n\}$$
and $$p(x_1,t_1,...,x_k,t_k|x_{k+1},t_{k+1},...,x_{n},t_n)=\frac{\partial ^k}{\partial x_1...\partial x_k}\mathbb P\{X_{t_1}\leq x_1,...,X_{t_k}\leq x_k\mid X_{t_{k+1}}=x_{k+1},...,X_n=t_n\}.$$
I want to prove that prove $$p(t_1,x_1,t_2,x_2|t_3,x_3)=p(t_1,x_1|t_2,x_2,x_3,t_3)p(t_2,x_2|t_3,x_3).$$
Attempts
$$p(t_1,x_1,t_2,x_2|t_3,x_3)=\frac{\partial ^2}{\partial x_1\partial x_2}\mathbb P\{X_{t_1}\leq x_1,X_{t_2}\leq x_2\mid X_3=x_3\}=\frac{\partial ^2}{\partial x_1\partial x_2}\int_{-\infty }^{x_2}\mathbb P\{X_{t_1}\leq x_1\mid X_{t_2}=u,X_{t_3}=x_3\}\mu_{X_{t_2}}(du)=\frac{\partial ^2}{\partial x_1\partial x_2}\int_{-\infty }^{x_2}\mathbb P\{X_1\leq x_1\mid X_{t_2}=u,X_{t_3}=x_3\}p(u,t_2)du$$ $$=\frac{\partial }{\partial x_1}\mathbb P\{X_{t_1}\leq x_1\mid X_{t_2}=x_2,X_{t_3}=x_3\}p(x_3,t_3)=p(t_1,x_1|t_2,x_2,t_3,x_3)p(t_2,x_2),$$ which is not what I want. Where is my mistakes ?