
theorem for a,b be Nat, c be non zero Nat holds
  (a mod c)*(b mod c) < c iff (a*b) mod c = (a mod c)*(b mod c)
  proof
    let a,b be Nat, c be non zero Nat;
    (a mod c)*(b mod c) < c implies (a*b) mod c = (a mod c)*(b mod c)
    proof
      assume (a mod c)*(b mod c) < c; then
      ((a mod c)*(b mod c)) mod c = (a mod c)*(b mod c) by NAT_D:24;
      hence thesis by NAT_D:67;
    end;
    hence thesis by INT_1:58;
  end;
