
theorem Th14x:
for F being Field
for n being Ordinal
for p being Monomial of n,F holds LC p = coefficient p & Lt p = term p
proof
let F be Field, n be Ordinal, p be Monomial of n,F;
H: Lt p is Element of Bags n &
   term(p) is Element of Bags n by PRE_POLY:def 12;
   field(BagOrder n) = Bags n by ORDERS_1:12; then
A: BagOrder n linearly_orders Support(p) by ORDERS_1:37,ORDERS_1:38;
per cases by POLYNOM7:def 5;
suppose p.term(p) <> 0.F;
  then AS: term(p) in Support p by H,POLYNOM1:def 4;
  then B: Support p = {term(p)} by POLYNOM7:7; then
  C: rng SgmX(BagOrder n,Support p) = {term(p)} by A,PRE_POLY:def 2;
  F: card Support p = 1 by B,CARD_1:30; then
  D: len SgmX(BagOrder n,Support p) = 1 by A,PRE_POLY:11;
  E: SgmX(BagOrder n,Support p) = <*term(p)*> by F,C,A,PRE_POLY:11,FINSEQ_1:39;
  p <> 0_(n,F) by AS,YY; then
  Lt p = SgmX(BagOrder n,Support p).1 by D,defLT
      .= term(p) by E;
  hence thesis by POLYNOM7:def 6;
  end;
suppose J: Support p = {} & term(p) = EmptyBag n; then
  K: LC p = 0.F by H,POLYNOM1:def 4; then
  L: LC p = p.(term(p)) by J,POLYNOM1:def 4
         .= coefficient(p) by POLYNOM7:def 6; then
  p = Monom(0.F,term(p)) by K,POLYNOM7:11; then
  M: term(p) = EmptyBag n by POLYNOM7:8;
  p = 0_(n,F) by J,YY;
  hence thesis by M,L,defLT;
  end;
end;
