theorem Th161:
  for R being Abelian right_zeroed add-associative
        right_complementable non empty addLoopStr,
      a, b being Element of R, i being Element of INT.Ring
  holds (Int-mult-left(R)).(i,a+b)
  = (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(i,b)
  proof
    let R be Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
    a, b be Element of R, i be Element of INT.Ring;
    reconsider ii = i as Element of INT;
    per cases;
    suppose 0 <= i; then
      reconsider i1=i as Element of NAT by INT_1:3;
      thus (Int-mult-left(R)).(i,a+b)
      = (Nat-mult-left(R)).(i1,a+b) by Def23
      .= i1*a + i1*b by Th160
      .= (Int-mult-left(R)).(i,a) + (Nat-mult-left(R)).(i1,b) by Def23
      .= (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(i,b) by Def23;
    end;
    suppose A1: 0 > i; then
      reconsider i1=-ii as Element of NAT by INT_1:3;
      (a+b) + ( (-a) + (-b)) = b + a + (-a) + (-b) by RLVECT_1:def 3
      .= b + (a + (-a)) + (-b) by RLVECT_1:def 3
      .= b + 0.R + (-b) by RLVECT_1:5
      .= b + (-b) by RLVECT_1:4
      .= 0.R by RLVECT_1:5; then
  A2: -(a+b) = (-a) + (-b) by RLVECT_1:6;
S1:   (i1)*(-a) = (Nat-mult-left(R)).(-i,-a)
               .= (Int-mult-left(R)).(i,a) by A1,Def23;
S2:   i1*(-b) = (Nat-mult-left(R)).(-i,-b);
      thus (Int-mult-left(R)).(i,a+b) = (Nat-mult-left(R)).(-i,-(a+b))
      by Def23,A1
      .= i1*(-a) + i1*(-b) by A2,Th160
      .= (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(i,b)
        by A1,Def23,S1,S2;
    end;
  end;
