reserve k for Nat;

theorem Th3:
  for a,b be Integer, n be Nat holds (a*b) mod n = (a*(b mod n)) mod n
proof
  let a,b be Integer;
  let n be Nat;
  per cases;
  suppose n > 0;
  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 thesis by NAT_D:64;
  end;
  suppose
A1: n = 0;
    hence (a*b) mod n = 0 by INT_1:def 10
    .= (a*(b mod n)) mod n by A1,INT_1:def 10;
  end;
end;
