Prove the following statement.

For every positive integer n$,$ if

A$1,$ A$2,$ A$3,...,$ An

and B are any sets, then

$$ni $=1($Ai $$ B$)=$

$$ni $=1$Ai

$$ B$.$

Proof: Suppose

A$1,$ A$2,$ A$3,...,$ An

and B are any sets. To prove that

$$ni $=1($Ai $$ B$)=$

$$ni $=1$Ai

$$ B$,$

we must show that

$$ni $=1($Ai $$ B$)$

$$ni $=1$Ai

$$ B$,$

and that

$$ni $=1$Ai

$$ B $$ni $=1($Ai $$ B$).$

Proof that

$$ni $=1($Ai $$ B$)$

$$ni $=1$Ai

$$ B:

$1.$ Suppose that

x in $$ni $=1($Ai $$ B$).$

$2.$ Then

x in Ai $$ B

for

$---$Select$---$

i $=1,2,...,$ n

by definition of

$---$Select$---$

$.$

$3.$ It follows that

x in Ai

for

$---$Select$---$

i $=1,2,...,$ n

$---$Select$---$

x

$?$

B

by definition of

$---$Select$---$

$.$

$4.$ Hence,

x in

$$ni $=1$Ai

by definition of

$---$Select$---$

$,$

$---$Select$---$

x

$?$

B

by step $3.$

Therefore,

x in

$$ni $=1$Ai

$$ B

by definition of

$---$Select$---$

$.$

Proof that

$$ni $=1$Ai

$$ B $$ni $=1($Ai $$ B$)$:

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