
theorem Z2a:
for R being non degenerated Ring
for n being Ordinal
for p being Polynomial of n,R holds LM p = 0_(n,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 B: Support p = {} by YY;
   Support(LM p) c= Support p by YZ;
   then Support(LM p) = {} by B;
   hence LM p = 0_(n,R) by YY;
   end;
now assume LM p = 0_(n,R);
  then Support(LM p) = {} by YY;
  then 0.R = (LM p).(Lt p) by H,POLYNOM1:def 4 .= LC p by lemY;
  hence p = 0_(n,R) by Y0;
  end;
hence thesis by A;
end;
