
theorem ZZZ:
for F being Field
for m being Ordinal st m in card(nonConstantPolys F)
for p being constant Polynomial of F
holds Lt Poly(m,p) = EmptyBag card(nonConstantPolys F) & LC Poly(m,p) = p.0
proof
let F be Field, m be Ordinal;
assume AS: m in card(nonConstantPolys F);
let p be constant Polynomial of F;
set q = Poly(m,p), n = card(nonConstantPolys F);
reconsider p1 = p as Element of the carrier of Polynom-Ring F
   by POLYNOM3:def 10;
I: p1 is constant by RATFUNC1:def 2;
   field(BagOrder n) = Bags n by ORDERS_1:12; then
J: BagOrder n linearly_orders Support q by ORDERS_1:37,ORDERS_1:38; then
B: rng SgmX(BagOrder n,Support q) = Support q by PRE_POLY:def 2;
per cases;
suppose A: q <> 0_(n,F);
  consider a being Element of F such that H: p1 = a|F by I,RING_4:20;
  C: q = a|(n,F) by H,AS,XYZbb;
  D: Support q = {EmptyBag n}
     proof
     E: now let o be object;
        assume D2: o in Support q;
        then reconsider b = o as bag of n;
        now assume o <> EmptyBag n; then
           D3: support b <> {} by PRE_POLY:81;
           per cases;
           suppose D4: support b = {m};
             m in {m} by TARSKI:def 1;
             then D5: b.m <> 0 by D4,PRE_POLY:def 7;
             q.b = p.(b.m) by D4,defPg .= 0.F by D5,H,Th28;
             hence contradiction by D2,POLYNOM1:def 4;
             end;
           suppose support b <> {m};
             then q.b = 0.F by D3,defPg;
             hence contradiction by D2,POLYNOM1:def 4;
             end;
           end;
        hence o in {EmptyBag n} by TARSKI:def 1;
        end;
     now let o be object;
        assume o in {EmptyBag n}; then
        D1: o = EmptyBag n by TARSKI:def 1;
        D2: q.(EmptyBag n) = p.0 by defPg .= a by H,Th28;
        a <> 0.F by A,C,POLYNOM7:19;
        hence o in Support q by D1,D2,POLYNOM1:def 4;
        end;
     hence thesis by E,TARSKI:2;
     end; then
  E: card Support q = 1 by CARD_1:30; then
  C: len SgmX(BagOrder n,Support q) = 1 by J,PRE_POLY:11;
  SgmX(BagOrder n,Support q)
        = <* EmptyBag n *> by D,B,E,J,PRE_POLY:11,FINSEQ_1:39;
  hence Lt Poly(m,p) = <* EmptyBag n *>.1 by C,A,defLT
                    .= EmptyBag n;
  hence LC Poly(m,p) = p.0 by defPg;
  end;
suppose q = 0_(n,F);
  hence Lt Poly(m,p) = EmptyBag n by defLT;
  hence LC Poly(m,p) = p.0 by defPg;
  end;
end;
