Consider the following Cauchy problem for heat equation: $u_t - \Delta_xu=f(x,t), x \in \mathbb{R}_n, t>0; u|_{t=0}=\phi(x), x \in \mathbb{R}_n$ where $u \in C^2(\{x \in \mathbb{R}_n; t>0\}) \cap C(\{x \in \mathbb{R}_n, t \geq 0\})$. It is proven that when $f(x,t) \equiv 0$ and $\phi(x)$ is bounded on $\mathbb{R}_n$ the Poisson integral solves the problem and $\inf_{\mathbb{R}_n} \phi(x) \leq u(x,t) \leq \sup_{\mathbb{R}_n}\phi(x)$.
Now the set $B$ is defined as the set of all functions $g(x,t)$ defined on $\{x \in \mathbb{R}_n, t\geq0\}$ that are bounded on $\{x \in \mathbb{R}_n, 0 \leq t \leq T\}$ for any $T>0$. It is proven that the solution of the general problem from $B$ is unique.
Now what I don't understand. According to my textbook, the uniqueness theorem implies that the solution $u(x,t) \in B$ has to be the Poisson integral. Moreover, it states that $\inf_{\mathbb{R}_n} \phi(x) \leq u(x,t) \leq \sup_{\mathbb{R}_n}\phi(x)$ holds here too. While the information up to here should be sufficient to derive the maximum principle, later on the textbook finds the following expression for $u \in B, f \in C(B)$: $u(x,t) = \int_{\mathbb{R}_n} {K(x-y,t)\phi(y)dy} + \int_0^t { \dfrac{d\tau} {(2\sqrt{\pi (t-\tau)})^n}} \int_{\mathbb{R}_n} {K(x-y,t-\tau)f(y,\tau)dy}$ where $K(z,t) \equiv \dfrac{1}{(2\sqrt{\pi t})^n}e^{-\dfrac{|z|^2}{4t}}$. The first one is what is defined as Poisson integral. But even so, I can't derive the principle. Any hints?
Edit1: a known fact is that $\int_{R_n} {K(x-y,t)dy} = 1$.