Let
x ∈ {A ∩ (B − C)}
x ∈ A and x ∈ B and x ∉ C
(x ∈ A and x ∈ B) and (x ∈ A and x ∉ C)
(x ∈ A ∩ B − A ∩ C)
A ∩ (B ∩ C) ⊆ (A ∩ B) − (A ∩ C) …(i)
Again, let
y ∈ (A ∩ B) ∩ (A − C)
y ∈ A and (y ∈ B and y ∉ C)
y ∈ A and y ∈ B − C
y ∈ {A ∩ (B − C)}
(A ∩ B) − (A ∩ C) ⊆ A ∩ (B − C) …(ii)
From equation (i) and (ii),
we get
A ∩ (B − C) = (A ∩ B) - (A ∩ C)