Note that this principle is less general than the principle of substitutivity for equalities, because it only applies to expressions that are built from integers and certain operations (note that division is not one of these operations). But it still lets us prove analogues of our above examples (51) and (52): • If n is any integer, and if a, b, c, d, e, c′ are integers such that c ≡ c′ mod n, then the principle of substitutivity for congruences says that we can replace c by c′ in the expression a (b − (c + d) e), and the value of the resulting expression a (b − (c′ + d) e) will be congruent to the value of a (b − (c + d) e) modulo n; that is, we have a (b − (c + d) e) ≡ a b − c′ + $$
•Ifnisanyinteger,andifa,b,c,a′areintegerssuchthata≡a′modn,then(a−b)(a+b)≡a′−(a+b)modn,(54)becausetheprincipleofsubstitutivityallowsustoreplacethefirstaappear−ingintheexpression(a−b)(a+b)byana′.(Wecanalsoreplacethesecondabya′,ofcourse.)Weshallnotprovetheprincipleofsubstitutivityforcongruences,sincewehavenotformalizedit(afterall,wehavenotdefinedwhatan“expression”is).Butweshallprovethespecificcongruences(53)and(54)usingProposition2.21andProposition2.12;thewayinwhichweprovethesecongruencesissymptomatic:Everycongruenceobtainedfromtheprincipleofsubstitutivityforcongruencescanbeproveninamannerlikethese.Thus,wehopethattheproofsof(53)and(54)givenbelowserveastemplateswhichcaneasilybeadaptedtoanyothersituationinwhichanapplicationoftheprincipleofsubstitutivityforcongruencesneedstobejustified. 
