= Solution
Write $a=P(A_1)=P(A_2)$. The complements are <decreasing events>. The <Harris-FKG inequality> for a <product measure> gives
$$
P(A_1^c\cap A_2^c)\ge P(A_1^c)P(A_2^c)=(1-a)^2.
$$
The same correlation inequality for two <decreasing events> follows from that for their increasing complements, since their <covariance> is unchanged. Thus
$$
1-P(A_1\cup A_2)\ge(1-a)^2.
$$
Both $1-a$ and the square root are nonnegative, so rearranging gives the <square-root bound for increasing events>:
$$
\boxed{P(A_1)\ge1-\sqrt{1-P(A_1\cup A_2)}.}
$$
No independence between the events themselves is asserted; independence is the coordinate property of the underlying <product measure>.
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