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Solution

If A is a proper subset of B as given here , then if x is in A then x is in B.

We can show the equality of sets by mutual inclusion of the sets.

Let x be in A. Since A is a proper subset of B, x is also in B.
Since x is in A and B both that means that x is in the intersection of A and B.

or x belongs to A intersection B or

A is a subset of A intersection of B because that x is also in A already.
Therefore, A is contained in A intersect B or A is a subset of A intersection B .......(i)

Let x be in A intersect B. Then x is in A and x is in B.
Since x is in A, x is contained in A.
Therefore, A intersect B is contained in A.

or A intersection B is a subset of A. ........(ii)

So by (i) and (ii) , both sets A and A intersection B are equal,

or A = A intersection B

Proved.

Since the two sets (A intersect B and A) are contained within each other, the sets are equal.