Equivalency statement wrt Internal Direct Product

Question

I am trying to show that if $G=H_1\times H_2 \times \dots \times H_n$ then the following are equivalent: $(H_1\times \dots \times H_{i-1})\cap H_i = \{e\} \iff H_i\cap H_j = \{e\}$ I have the $\impliedby$ part down, but I am having a hard time formalizing the reverse. I would appreciate any help.

Answer 1

Suppose $\;1\neq x\in H_i\cap H_j\;,\;\;i<j\;$ , so

$$1\neq x=1\cdot 1\cdot\ldots\cdot \overbrace{x}^{i}\in (H_1\cdot\ldots\cdot H_i)\cap H_j\;\;\;\leftarrow\text{contradiction}$$