I'm reading "Introduction to Quantum Field Theory for Mathematicians, Lecture Notes, sourav Chatterjee (https://souravchatterjee.su.domains//qft-lectures-combined.pdf), p.21 and stuck at understanding some equality.
In his lecture note, p.7, he states Stone's Theorem as
There is a one-to-one correspondence between one parameter strongly continuous unitary groups of operators on $\mathcal{H}$ (Hilbert space) and self-adjoint operators on $\mathcal{H}$. Given $U$, the corresponding self-adjoint operator $A$ is defined as $$ Ax = \lim_{t\to 0}\frac{U(t)x-x}{it}$$ with $\mathcal{D}(A) = \{x : \text{the above limit exists} \}$.
And he propose Postulate 5 of quantum mechanics :
P5. If the system is not affected by external influences then its state evolves in time as $\psi_t = U(t)\psi$ for some strongly continuous unitary group $U$ that only depends on the system(and not on the state)
By Stone's theorem, there exsists a unique self-adjoint operator $H$ corresponding to $U(t)$. It is called the 'Hamiltonian'.
And in his note, p.21, he saids that
"Suppose that the state of a single particle evolves according to the unitary group $(U(t))_{t\in \mathbb{R}}$. Then the time evolution on $\mathcal{H}^{\otimes n}$ of $n$ non-interacting particles, also denoted by $U(t)$, is defined as
$$ U(t)(\psi_1 \otimes \cdots \otimes \psi_n) := (U(t)\psi_1)\otimes(U(t)\psi_2)\otimes \cdots \otimes (U(t)\psi_n)$$
and extended by linearity."
Then, he calculated an action of the hamiltonian on $\psi_1 \otimes \cdots \otimes \psi_n$ by
My question is, why the underlined equality is true?
Can anyone help?