
theorem Th15:
  for L being add-associative right_zeroed right_complementable
  left-distributive non empty doubleLoopStr for p being Polynomial of L for x
  being Element of L holds - (x * p) = (-x) * p
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  non empty doubleLoopStr, p be Polynomial of L;
  let x be Element of L;
  set f = - (x * p), g = (-x) * p;
A1: now
    let i9 be object;
    assume i9 in dom f;
    then reconsider i = i9 as Element of NAT;
    f.i = -((x*p).i) by BHSP_1:44
      .= -(x*p.i) by POLYNOM5:def 4
      .= (-x)*p.i by VECTSP_1:9
      .= g.i by POLYNOM5:def 4;
    hence f.i9 = g.i9;
  end;
  dom f = NAT by FUNCT_2:def 1
    .= dom g by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
