
theorem Th14z:
for F being Field
for m being Ordinal st m in card(nonConstantPolys F)
for p being Polynomial of F
holds LC Poly(m,LM p) = LC Poly(m,p) & Lt Poly(m,LM p) = Lt Poly(m,p)
proof
let F be Field, m be Ordinal;
assume AS: m in card(nonConstantPolys F);
let p be Polynomial of F;
set n = card(nonConstantPolys F);
thus LC Poly(m,LM p) = LC(LM p) by AS,Th14b
               .= (LM p).(len(LM p) -' 1) by RATFUNC1:def 6
               .= (LM p).(len p -' 1) by POLYNOM4:15
               .= p.(len p -' 1) by POLYNOM4:def 1
               .= LC p by RATFUNC1:def 6
               .= LC Poly(m,p) by Th14b,AS;
per cases;
suppose I0: p is constant;
  hence Lt Poly(m,p) = EmptyBag n by AS,ZZZ
                    .= Lt Poly(m,LM p) by AS,I0,ZZZ;
  end;
suppose I0: p is non constant;
  len(LM p) = len p by POLYNOM4:15; then
  H: deg(LM p) = len p - 1 by HURWITZ:def 2 .= deg p by HURWITZ:def 2;
  deg p > 0 by I0,RATFUNC1:def 2; then
  I1: LM p is non constant by H,RATFUNC1:def 2;
  set b1 = Lt Poly(m,p), b2 = Lt Poly(m,LM p);
  now let o be object;
    assume o in n;
    then reconsider i  = o as Ordinal;
    per cases;
    suppose I4: i = m;
      then b1.i = deg p by I0,AS,XYZaa .= b2.i by H,I4,I1,AS,XYZaa;
      hence b1.o = b2.o;
      end;
    suppose I4: i <> m;
      then b1.i = 0 by I0,AS,XYZaa .= b2.i by I4,I1,AS,XYZaa;
      hence b1.o = b2.o;
      end;
    end;
  hence thesis by PBOOLE:def 10;
  end;
end;
