reserve n,k,b for Nat, i for Integer;

theorem
  for i,j,k,l being Integer st i mod l = j mod l holds
  (k + i) mod l = (k + j) mod l
  proof
    let i,j,k,l be Integer;
    assume A1: i mod l = j mod l;
    thus (k + i) mod l = ((k mod l)+(i mod l)) mod l by NAT_D:66
    .= (k + j) mod l by A1,NAT_D:66;
  end;
