theorem Th154:
  for R being add-associative right_zeroed right_complementable
  non empty addLoopStr,
  i be Element of INT.Ring holds
  (Int-mult-left(R)).(i,0.R) = 0.R
  proof
    let R be add-associative right_zeroed right_complementable
    non empty addLoopStr,
    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,0.R) = (Nat-mult-left(R)).(i1,0.R) by Def23
      .=0.R by Th153;
    end;
    suppose A1: 0 > i; then
      reconsider i1=-ii as Element of NAT by INT_1:3;
      thus (Int-mult-left(R)).(i,0.R) = (Nat-mult-left(R)).(-i,-(0.R))
      by Def23,A1
      .= (Nat-mult-left(R)).(i1,0.R) by RLVECT_1:12
      .=0.R by Th153;
    end;
  end;
