xy + yz + zx′ = xy + zx′ (2.21a)

(x + y)(y + z)(z + x′) = (x+ y)(z +x′) (2.21b)

(a) Proof of the main theorem: First, multiply the middle term yz with to yield

LHS = xy + (x + x′)yz + zx′ = xy + xyz + x′yz x′z

= xy(1 + z) + x′z(1 + y) = xy + xz = RHS

(b) Proof of the complementary theorem: To prove the complement of the consensus theorem, as before, consider the LHS term:

LHS = (x+ y)(y +z)(z + x′) = (xy + xz + y + yz)(z + x′)

= xyz + xz + yz + yx′ + xz + x′yz

= xz + yz + yx′

Now, we expand RHS, which yields

RHS = (x+y)(z+x′) = xz + yz +x′y

LHS = RHS

**Extended Form I of the Consensus Theorem**

This theorem states that:

x′y′ + y′z′ + z′x = x′y′ + z′x

Proof: We have the LHS given by

LHS = x′y′ + y′z′ + z′x

Adding x and x′ to the middle term yields

x′y′ + x′y′z′ + xy′z′ +z′x = x′y′(1+ z′) + zx′(1+ y′) = x′y′ + z′x = RHS

Complementary of Extended Form I

(x′ + y′)(y′ + z′)(z′ + x) = (x′ + y′)(z′ + x)

**Extended Form II of the Consensus Theorem**

This theorem states that:

xy +yzw + zx′ = xy + zx

Proof: Proof follows the same steps as given above. It can also be noticed that the middle term can contain any number of terms; all of them will get eliminated as illustrated below:

xy + yzwabcd..... + zx′ = xy + zx′