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

theorem Th14:
  for i,j being Integer holds (i mod j) mod j = i mod j
  proof
    let i,j be Integer;
    per cases;
    suppose j<>0;
      hence (i mod j) mod j = (i - (i div j) * j) mod j by INT_1:def 10
      .= (i + j * (-(i div j) qua Integer)) mod j
      .= i mod j by NAT_D:61;
    end;
    suppose A1: j=0;
      hence (i mod j) mod j = 0 by INT_1:def 10 .= i mod j by A1,INT_1:def 10;
    end;
  end;
