prove that – (A∩B)X(C∩D) = (AXC)∩(BXD)

Prob. If A, B, C, D are any four sets then prove that –
(A∩B)X(C∩D) = (AXC)∩(BXD)

Solution:
Consider(x,y)
(x,y)∈(A∩B)×(C∩D)
x∈(A∩B) ∧ y∈(C∩D)
(x∈A and x∈B) ∧ (y∈C and y∈D)
(x∈A ∧ y∈C) and (x∈B ∧ y∈D)
(x,y)∈(A∧C) and (x,y)∈(B∧D)
(x,y)∈((A∧C) and (B∧D))
(x,y)∈((A×C) ∩ (B×D))

(A×C) ∩ (B×D)