 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;
 reserve R for Ring;
 reserve V, X, Y for LeftMod of R;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a for Element of R;
 reserve V1, V2, V3 for Subset of V;
 reserve x for set;
 reserve W, W1, W2 for Submodule of V;
 reserve w, w1, w2 for Vector of W;
 reserve D for non empty set;
 reserve d1 for Element of D;
 reserve A for BinOp of D;
 reserve M for Function of [:the carrier of R,D:],D;
reserve B,C for Coset of W;
 reserve V for LeftMod of R;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a, a1, a2 for Element of R;
 reserve X, Y, y, y1, y2 for set;
 reserve C for Coset of W;
 reserve C1 for Coset of W1;
 reserve C2 for Coset of W2;
reserve A1,A2,B for Element of Submodules(V);

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;
