theorem Th158:
  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
  st i in NAT & not j in NAT
  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 jj = j, ii = i, ij = i+j as Element of INT;
    assume A1: i in NAT & not j in NAT; then
    reconsider i1=i as Element of NAT;
A2: jj < 0 by A1,INT_1:3; then
    reconsider j1=-jj as Element of NAT by INT_1:3;
    per cases;
    suppose A4: j1 <= i1;
      reconsider k1=i1-j1 as Element of NAT by A4,INT_1:5;
      set IT = Int-mult-left(R);
W1:   (Int-mult-left(R)).(jj,a) = (Nat-mult-left(R)).(-j,-a) by A2,Def23;
      thus (Int-mult-left(R)).(i+j,a) = (Nat-mult-left(R)).(k1,a) by Def23
      .= (Nat-mult-left(R)).(i1,a) - (Nat-mult-left(R)).(j1,a) by Th156,A4
      .=(Nat-mult-left(R)).(i1,a) + (Nat-mult-left(R)).(j1,-a) by Th157
      .=(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 W1;
    end;
    suppose A5: j1 > i1; then
      reconsider k1=j1-i1 as Element of NAT by INT_1:5;
      A6: i1-j1 < 0 by A5,XREAL_1:49;
Z1:   (Int-mult-left(R)).(j,a) = (Nat-mult-left(R)).(-j,-a) by Def23,A2;
      thus (Int-mult-left(R)).(i+j,a)
       = (Nat-mult-left(R)).(-(i+j),-a) by A6,Def23
      .= (Nat-mult-left(R)).(k1,-a)
      .= (Nat-mult-left(R)).(j1,-a) - (Nat-mult-left(R)).(i1,-a)
      by Th156,A5
      .=(Nat-mult-left(R)).(j1,-a) + (Nat-mult-left(R)).(i1,-(-a))
      by Th157
      .=(Nat-mult-left(R)).(j1,-a) + (Nat-mult-left(R)).(i1,a)
      by RLVECT_1:17
      .=(Int-mult-left(R)).(j,a) + (Int-mult-left(R)).(i,a)
      by Def23,Z1
      .=(Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a);
    end;
  end;
