
theorem Y0:
for R being non degenerated Ring
for n being Ordinal
for p being Polynomial of n,R holds LC p = 0.R iff p = 0_(n,R)
proof
let R be non degenerated Ring, n be Ordinal, p be Polynomial of n,R;
H: Lt p is Element of Bags n by PRE_POLY:def 12;
A: now assume p = 0_(n,R);
   then Support p = {} by YY;
   hence LC p = 0.R by H,POLYNOM1:def 4;
   end;
now assume LC p = 0.R;
   then B: not Lt p in Support p by POLYNOM1:def 4;
   field(BagOrder n) = Bags n by ORDERS_1:12; then
   K: BagOrder n linearly_orders Support p by ORDERS_1:37,ORDERS_1:38;
   now assume I: p <> 0_(n,R); then
     G: Lt p = SgmX(BagOrder n,Support p).(len SgmX(BagOrder n,Support p))
        by defLT;
     L: rng SgmX(BagOrder n,Support p) = Support p by K,PRE_POLY:def 2;
     Support p <> {} by I,YY;
     then M: 1 <= len SgmX(BagOrder n,Support p) by FINSEQ_1:20;
     dom SgmX(BagOrder n,Support p) = Seg(len SgmX(BagOrder n,Support p))
       by FINSEQ_1:def 3; then
     len SgmX(BagOrder n,Support p) in dom SgmX(BagOrder n,Support p)
       by M;
     hence contradiction by B,L,G,FUNCT_1:def 3;
     end;
   hence p = 0_(n,R);
   end;
hence thesis by A;
end;
