
theorem Th12:
for F being Field
for p being non constant Element of the carrier of Polynom-Ring F
for m being Ordinal st m in card(nonConstantPolys F)
holds Poly(m,anpoly(LC p,deg p)) = LM Poly(m,p)
proof
let F be Field, p be non constant Element of the carrier of Polynom-Ring F;
let m be Ordinal;
set n = card(nonConstantPolys F);
assume AS: m in n;
deg p > 0 by RING_4:def 4; then
len p - 1 > 0 by HURWITZ:def 2; then
len p - 1 = len p -' 1 by XREAL_0:def 2; then
deg p = len p -' 1 by HURWITZ:def 2;
then anpoly(LC p,deg p)
   = anpoly(p.(deg p),deg p) by RATFUNC1:def 6
  .= LM p by FIELD_1:11;
hence thesis by AS,Th14y;
end;
