I assume A - B is the complement of B in A, i.e. all elements of A that do not belong to B.
To prove equality between two sets, you have to show they are subsets of one another. To prove one set is a subset of another set, you have to show that any element in the first set belongs to the second set.
So, to show that A - B = A ∩ (C - B), you have to prove
• A - B ⊆ A ∩ (C - B)
• A ∩ (C - B) ⊆ A - B
Left to right:
Let x ∈ A - B.
By definition of set complement, x ∈ A and x ∉ B.
Since A ⊆ C, x ∈ C.
So x ∈ C - B, and by definition of set intersection, x ∈ A ∩ (C - B).
This proves A - B ⊆ A ∩ (C - B).
Right to left:
Let x ∈ A ∩ (C - B).
By definition of set intersection, x ∈ A and x ∈ C - B.
By definition of set complement, x ∈ C and x ∉ B.
Again by definition of complement, x ∈ A and x ∉ B means that x ∈ A - B.
This proves A ∩ (C - B) ⊆ A - B.
Both sets are subsets of one another, so A - B = A ∩ (C - B).