let a, b be Integer; :: thesis: for n being Nat holds (a * b) mod n = (a * (b mod n)) mod n
let n be Nat; :: thesis: (a * b) mod n = (a * (b mod n)) mod n

per cases ( n > 0 or n = 0 ) ;

suppose n > 0 ; :: thesis: (a * b) mod n = (a * (b mod n)) mod n

then (b mod n) + ((b div n) * n) = (b - ((b div n) * n)) + ((b div n) * n) by INT_1:def 10
.= b + 0 ;
then (a * b) - (a * (b mod n)) = (a * (b div n)) * n ;
then n divides (a * b) - (a * (b mod n)) by INT_1:def 3;
then a * b,a * (b mod n) are_congruent_mod n by INT_2:15;
hence (a * b) mod n = (a * (b mod n)) mod n by NAT_D:64; :: thesis: verum

end;

suppose A1: n = 0 ; :: thesis: (a * b) mod n = (a * (b mod n)) mod n

hence (a * b) mod n = 0 by INT_1:def 10
.= (a * (b mod n)) mod n by A1, INT_1:def 10 ;
:: thesis: verum

end;

end;