
theorem Th14xy:
for F being Field
for m being Ordinal
for p being Polynomial of F holds Poly(m,-p) = - Poly(m,p)
proof
let F be Field, m be Ordinal, p be Polynomial of F;
set r1 = Poly(m,p), r2 = Poly(m,-p), n = card(nonConstantPolys F);
I: dom r2 = Bags n by FUNCT_2:def 1 .= dom(-r1) by FUNCT_2:def 1;
now let o be object;
  assume   o in dom r2;
  then reconsider b = o as bag of n;
  per cases;
  suppose support b = {};
    then for i being object st i in n holds b.i = {} by PRE_POLY:def 7;
    then H: b = EmptyBag n by PBOOLE:6;
    r2.b = (-p).0 by H,defPg
        .= -(p.0) by BHSP_1:44
        .= -(r1.b) by H,defPg
        .= (-r1).b by POLYNOM1:17;
    hence r2.o = (-r1).o;
    end;
  suppose H: support b = {m};
    then r2.b = (-p).(b.m) by defPg
             .= - (p.(b.m)) by BHSP_1:44
             .= -(r1.b) by H,defPg
             .= (-r1).b by POLYNOM1:17;
    hence r2.o = (-r1).o;
    end;
  suppose H: support b <> {} & support b <> {m};
    then r2.b = - 0.F by defPg
             .= -(r1.b) by H,defPg
             .= (-r1).b by POLYNOM1:17;
    hence r2.o = (-r1).o;
    end;
  end;
hence thesis by I;
end;
