
theorem bij3:
for F being Field
for m1,m2 being Ordinal
st m1 in card(nonConstantPolys F) & m2 in card(nonConstantPolys F)
for p1,p2 being non constant Polynomial of F
st Poly(m1,p1) = Poly(m2,p2) holds m1 = m2 & p1 = p2
proof
let F be Field, m1,m2 be Ordinal;
assume A: m1 in card(nonConstantPolys F) & m2 in card(nonConstantPolys F);
let p1,p2 be non constant Polynomial of F;
set n = card(nonConstantPolys F);
assume B: Poly(m1,p1) = Poly(m2,p2);
H: Support Poly(m1,p1)
    c= {EmptyBag n} \/ {b where b is bag of n : support b = {m1}} &
   Support Poly(m2,p2)
    c= {EmptyBag n} \/ {b where b is bag of n : support b = {m2}}by Th14c;
ex b being bag of n st
    b in Support Poly(m1,p1) & b in Support Poly(m2,p2) & b <> EmptyBag n
   proof
   p1 <> 0_.(F); then
   B0: Poly(m1,p1) <> 0_(n,F) by A,pZero;
   deg p1 > 0 by RATFUNC1:def 2; then
   p1 is non constant Element of the carrier of Polynom-Ring F
     by RING_4:def 4,POLYNOM3:def 10; then
   B2: Support Poly(m1,p1) <> {EmptyBag n} by A,bij3a;
   now assume C0: not ex b being object st
         b in Support Poly(m1,p1) & b <> EmptyBag n;
     now let o be object;
       assume o in Support Poly(m1,p1);
       then o = EmptyBag n by C0;
       hence o in {EmptyBag n} by TARSKI:def 1;
       end; then
     Support Poly(m1,p1) c= {EmptyBag n};
     hence contradiction by B0,YY,B2,lemSet;
     end; then
   consider b being object such that
   B3: b in Support Poly(m1,p1) & b <> EmptyBag n;
   reconsider b as bag of n by B3;
   thus thesis by B3,B;
   end;
then consider b being bag of n such that
D0: b in Support Poly(m1,p1) & b in Support Poly(m2,p2) & b <> EmptyBag n;
b in {EmptyBag n} \/ {b where b is bag of n : support b = {m1}} &
  not b in {EmptyBag n} by D0,H,TARSKI:def 1;
then b in {b where b is bag of n : support b = {m1}} by XBOOLE_0:def 3;
then consider b1 being bag of n such that
D1: b = b1 & support b1 = {m1};
b in {EmptyBag n} \/ {b where b is bag of n : support b = {m2}} &
  not b in {EmptyBag n} by D0,H,TARSKI:def 1;
then b in {b where b is bag of n : support b = {m2}} by XBOOLE_0:def 3;
then consider b2 being bag of n such that
D2: b = b2 & support b2 = {m2};
m1 in {m2} by D1,D2,TARSKI:def 1;
hence X: m1 = m2 by TARSKI:def 1;
now let i be Element of NAT;
  per cases;
  suppose E: i = 0;
    hence p1.i = Poly(m2,p2).(EmptyBag n) by B,defPg
              .= p2.i by E,defPg;
    end;
  suppose C: i <> 0;
    for o being object st o in {m1} holds o in n by A,TARSKI:def 1;
    then reconsider S = {m1} as finite Subset of n by TARSKI:def 3;
    set b = (S,i)-bag;
    E: support b = S by C,UPROOTS:8;
    F: m1 in {m1} by TARSKI:def 1;
    hence p1.i = p1.(b.m1) by UPROOTS:7
              .= Poly(m2,p2).b by E,B,defPg
              .= p2.(b.m2) by E,X,defPg
              .= p2.i by F,UPROOTS:7,X;
    end;
  end;
hence thesis;
end;
