
theorem Th8:
  for X being set, L being right_zeroed left_add-cancelable
left-distributive non empty doubleLoopStr, p being Series of X,L holds 0.L *
  p = 0_(X,L)
proof
  let n be set, L be right_zeroed left_add-cancelable left-distributive non
  empty doubleLoopStr, p be Series of n,L;
  set op = 0.L * p;
A1: now
    let u be object;
    assume u in dom op;
    then reconsider u9 = u as bag of n;
    op.u9 = 0.L * p.u9 by POLYNOM7:def 9
      .= 0.L by BINOM:1;
    hence op.u = 0_(n,L).u by POLYNOM1:22;
  end;
  dom op = Bags n by FUNCT_2:def 1
    .= dom 0_(n,L) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
