Proof

If a R is invertible with inverse b and a' R' is invertible with inverse b', then (a,a') is invertible with inverse (b,b'):

(a,a') * (b,b') = (a * b,a' * b') = (1,1'),

and similarly for (b,b') * (a,a').

Conversely, if (a,b) is invertible with inverse (c,c'), then, by the same kind of equalities as in the previous part of the proof, it follows immediately that a is invertible with inverse c and a' is invertible with inverse c'.