
theorem Th5:
  for n being Ordinal, L being right_zeroed add-associative
right_complementable non empty doubleLoopStr, p being Polynomial of n,L holds
  Support(-p) = Support(p)
proof
  let n be Ordinal, L be right_zeroed add-associative right_complementable
  non empty doubleLoopStr, p be Polynomial of n,L;
A1: now
    let u be object;
    assume
A2: u in Support(p);
    then reconsider u9 = u as Element of Bags n;
A3: p.u9 <> 0.L by A2,POLYNOM1:def 4;
    now
      assume (-p).u9 = 0.L;
      then (-p).u9 = -(-(0.L)) by RLVECT_1:17;
      then -(p.u9) = -(-(0.L)) by POLYNOM1:17
        .= 0.L by RLVECT_1:17;
      then p.u9 = -(0.L) by RLVECT_1:17;
      hence contradiction by A3,RLVECT_1:12;
    end;
    hence u in Support(-p) by POLYNOM1:def 4;
  end;
  now
    let u be object;
    assume
A4: u in Support(-p);
    then reconsider u9 = u as Element of Bags n;
A5: (-p).u9 <> 0.L by A4,POLYNOM1:def 4;
    now
A6:   (-p).u9 = -(p.u9) by POLYNOM1:17;
      assume p.u9 = 0.L;
      hence contradiction by A5,A6,RLVECT_1:12;
    end;
    hence u in Support(p) by POLYNOM1:def 4;
  end;
  hence thesis by A1,TARSKI:2;
end;
