
theorem pZero:
for F being Field
for m being Ordinal st m in card(nonConstantPolys F)
for p being Polynomial of F
holds Poly(m,p) = 0_(card(nonConstantPolys F),F) iff p = 0_.(F)
proof
let F be Field, m be Ordinal;
assume K: m in card(nonConstantPolys F);
let p be Polynomial of F;
set n = card(nonConstantPolys F);
H: Support Poly(m,p) c= {EmptyBag n} \/
                        {b where b is bag of n : support b = {m}} by Th14c;
A: now assume AS: p = 0_.(F);
   now let o be object;
     assume B: o in Support Poly(m,p);
     per cases by H,B,XBOOLE_0:def 3;
     suppose o in {EmptyBag n};
       then o = EmptyBag n by TARSKI:def 1;
       then Poly(m,p).o = p.0 by defPg;
       hence contradiction by B,AS,POLYNOM1:def 4;
       end;
     suppose o in {b where b is bag of n : support b = {m}};
       then consider b being bag of n such that
       D: o = b & support b = {m};
       Poly(m,p).b = p.(b.m) by D,defPg;
       hence contradiction by D,B,AS,POLYNOM1:def 4;
       end;
     end;
   then Support Poly(m,p) = {} by XBOOLE_0:def 1;
   hence Poly(m,p) = 0_(n,F) by YY;
   end;
now assume AS: Poly(m,p) = 0_(n,F);
   now let i be Element of NAT;
     per cases;
     suppose i = 0;
       hence p.i = Poly(m,p).(EmptyBag n) by defPg .= 0.F by AS,POLYNOM1:22;
       end;
     suppose A: i <> 0;
       for o being object st o in {m} holds o in n by K,TARSKI:def 1;
       then reconsider S = {m} as finite Subset of n by TARSKI:def 3;
       set b = (S,i)-bag;
       B: support b = S by A,UPROOTS:8;
       m in {m} by TARSKI:def 1;
       hence p.i = p.(b.m) by UPROOTS:7
                .= Poly(m,p).b by B,defPg .= 0.F by AS,POLYNOM1:22;
       end;
     end;
   hence p = 0_.(F);
   end;
hence thesis by A;
end;
