theorem Th159:
  for R being Abelian right_zeroed add-associative
    right_complementable non empty addLoopStr,
  a being Element of R, i, j being Element of INT.Ring
  holds (Int-mult-left(R)).(i+j,a)
  = (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a)
  proof
    let R be Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
    a be Element of R, i, j be Element of INT.Ring;
    reconsider ii = i, jj = j as Element of INT;
    per cases;
    suppose A1: i in NAT & j in NAT; then
      reconsider i1=i as Element of NAT;
      reconsider j1=j as Element of NAT by A1;
      thus (Int-mult-left(R)).(i+j,a)
       = (Nat-mult-left(R)).(i1+j1,a) by Def23
      .= (i1+j1)*a
      .= i1*a + j1*a by BINOM:15
      .= (Int-mult-left(R)).(i,a) + (Nat-mult-left(R)).(j1,a) by Def23
      .= (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a) by Def23;
    end;
    suppose i in NAT & not j in NAT;
      hence (Int-mult-left(R)).(i+j,a) = (Int-mult-left(R)).(i,a)
      +(Int-mult-left(R)).(j,a) by Th158;
    end;
    suppose not i in NAT & j in NAT;
      hence (Int-mult-left(R)).(i+j,a)
      =(Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a) by Th158;
    end;
    suppose not i in NAT & not j in NAT; then
      A3: i < 0 & j < 0 by INT_1:3; then
      reconsider i1=-ii as Element of NAT by INT_1:3;
      reconsider j1=-jj as Element of NAT by A3,INT_1:3;
S1:   (Int-mult-left(R)).(i,a) = (Nat-mult-left(R)).(-i,-a) by A3,Def23
         .= i1*(-a);
S2:   (Nat-mult-left(R)).(-j,-a) = j1*(-a);
      thus (Int-mult-left(R)).(i+j,a) = (Nat-mult-left(R)).(-(i+j),-a)
         by Def23,A3
      .= (i1+j1)*(-a)
      .= i1*(-a) + j1*(-a) by BINOM:15
      .= (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a)
        by A3,Def23,S1,S2;
    end;
  end;
