
theorem bij3a:
for F being Field
for m being Ordinal st m in card(nonConstantPolys F)
for p being non zero Element of the carrier of Polynom-Ring F
holds Support Poly(m,p) = {EmptyBag card(nonConstantPolys F)} iff p is constant
proof
let F be Field, m be Ordinal;
assume A: m in card(nonConstantPolys F);
let p be non zero Element of the carrier of Polynom-Ring F;
set n = card(nonConstantPolys F);
B: now assume B1: Support Poly(m,p) = {EmptyBag n};
   now let b be bag of n;
     assume b <> EmptyBag n;
     then B2: not b in Support Poly(m,p) by B1,TARSKI:def 1;
     b is Element of Bags n by PRE_POLY:def 12;
     hence Poly(m,p).b = 0.F by B2,POLYNOM1:def 4;
     end;
   then Poly(m,p) is Constant by POLYNOM7:def 7;
   then consider a being Element of F such that
   C: Poly(m,p) = a|(card(nonConstantPolys F),F) by POLYNOM7:17;
   p = a|F by C,A,XYZbb;
   hence p is constant by RING_4:20;
   end;
now assume p is constant; then
   consider a being Element of F such that
   C: p = a|F by RING_4:20;
   D: Poly(m,p) = a|(n,F) by C,A,XYZbb;
   p <> 0_.(F); then
   a|(n,F) <> 0_(n,F) by A,D,pZero;
   hence Support Poly(m,p) = {EmptyBag n} by D,POLYNOM7:14;
   end;
hence thesis by B;
end;
