Idem for p 2 {\displaystyle p_{2}} . We define the product channel p 1 × p 2 {\displaystyle p_{1}\times p_{2}} as ∀ ( x 1 , x 2 ) ∈ ( X 1 , X 2 ) , ( y 1 , y 2 ) ∈ ( Y 1 , Y 2 ) , ( p 1 × p 2 ) ( ( y 1 , y 2 ) | ( x 1 , x 2 ) ) = p 1 ( y 1 | x 1 ) p 2 ( y 2 | x 2 ) {\displaystyle \forall (x_{1},x_{2})\in ({\mathcal {X}}_{1},{\mathcal {X}}_{2}),\;(y_{1},y_{2})\in ({\mathcal {Y}}_{1},{\mathcal {Y}}_{2}),\;(p_{1}\times p_{2})((y_{1},y_{2})|(x_{1},x_{2}))=p_{1}(y_{1}|x_{1})p_{2}(y_{2}|x_{2})}