
theorem Th15:
  for L being add-associative right_zeroed right_complementable
distributive non empty doubleLoopStr, x being Element of Polynom-Ring L, p be
  sequence of L st x = p holds -x = -p
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr, x being Element of Polynom-Ring L, p be sequence of L;
  assume
A1: x = p;
  reconsider x9=x as Polynomial of L by POLYNOM3:def 10;
A2: now
    let i be object;
    assume
A3: i in NAT;
    then reconsider i9=i as Element of NAT;
    thus (p-p).i = p.i9 - p.i9 by POLYNOM3:27
      .= 0.L by RLVECT_1:15
      .= (0_.L).i by A3,FUNCOP_1:7;
  end;
  reconsider mx=-x9 as Element of Polynom-Ring L by POLYNOM3:def 10;
A4: p-p = 0_.L by A2;
  x+mx = x9+-x9 by POLYNOM3:def 10
    .=0.(Polynom-Ring L) by A1,A4,POLYNOM3:def 10;
  then x = -mx by RLVECT_1:6;
  hence thesis by A1,RLVECT_1:17;
end;
