
theorem YY:
for R being non degenerated Ring
for n being Ordinal
for p being Polynomial of n,R holds p = 0_(n,R) iff Support p = {}
proof
let R be non degenerated Ring, n be Ordinal, p be Polynomial of n,R;
A: now assume B: p = 0_(n,R);
   now let o be object;
     assume C: o in Support p;
     then reconsider b1 = o as Element of Bags n;
     p.b1 <> 0.R by C,POLYNOM1:def 4;
     hence contradiction by B,POLYNOM1:22;
     end;
   hence Support p = {} by XBOOLE_0:def 1;
   end;
now assume Support p = {};
   then p = (Bags n) --> 0.R by POLYNOM1:def 4;
   hence p = 0_(n,R) by POLYNOM1:def 8;
   end;
hence thesis by A;
end;
