Conditional variance and expectation of random variables

I am trying to show whether the following statement is true or not:

$E(X^2|A)E(1|A) \ge E(X|A)^2$

It is straightforward to prove this statement if $X$ was not conditioned on event $A$. Because, one can simply show

$E(X^2)E(1) = E(X^2)$ and $E(X^2)-E(X)^2 = E(X-E(X))^2 \ge 0$.

However, I am not sure whether the same logic would apply to the conditional expectations or not.

Many thanks!

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migrated from mathoverflow.netJul 10 '14 at 7:20

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Got something from the answer below? –  Did Jul 15 '14 at 17:38

The same argument works: expand the LHS of the inequality $$E((X-x)^2|A)\geqslant0,\qquad\text{with}\ x=E(X|A).$$