
theorem r59a:
for F being Field,
    S being non empty finite Subset of F
for a being Element of F, p being Ppoly of F,(S \/ {a})
for q being non constant Polynomial of F
st p = rpoly(1,a) *' q & not a in S holds q is Ppoly of F,S
proof
let F be Field, S be non empty finite Subset of F, a be Element of F;
let p be Ppoly of F,(S \/ {a}); let q be non constant Polynomial of F;
assume A: p = rpoly(1,a) *' q & not a in S;
B1: Roots rpoly(1,a) = {a} by RING_5:18;
B2: rpoly(1,a) is Ppoly of F by RING_5:51;
B6: now let b be Element of F;
    assume b is_a_root_of rpoly(1,a);
    then b in Roots rpoly(1,a) by POLYNOM5:def 10;
    then b = a by B1,TARSKI:def 1;
    hence multiplicity(rpoly(1,a),b) = 1 by RING_5:34;
    end;
B3: rpoly(1,a) is Ppoly of F,{a} by B1,B2,B6,ZZ1;
B8: S \/ {a} = Roots p by RING_5:63;
B10: p is Ppoly of F,(Roots p) by RING_5:63;
B9: a in Roots rpoly(1,a) by B1,TARSKI:def 1;
B7: for b be Element of F st b is_a_root_of rpoly(1,a) *' q
    holds multiplicity(rpoly(1,a) *' q,b) = 1 by A,B10,ZZ1;
C: S = Roots q
   proof
   C1: now let o be object;
       assume C2: o in S; then
       reconsider b = o as Element of F;
       b in S \/ {a} by C2,XBOOLE_0:def 3; then
       b in Roots p by RING_5:63; then
       C4: b is_a_root_of rpoly(1,a) *' q by A,POLYNOM5:def 10;
       not b in {a} by C2,A,TARSKI:def 1; then
       not b in Roots rpoly(1,a) by RING_5:18; then
       not b is_a_root_of rpoly(1,a) by POLYNOM5:def 10;
       then b is_a_root_of q by C4,ZZ1e;
       hence o in Roots q by POLYNOM5:def 10;
       end;
   now let o be object;
       assume C2: o in Roots q; then
       reconsider b = o as Element of F;
       C3: Roots q c= Roots p by A,ZZ1f;
       now assume b = a; then
         b in Roots rpoly(1,a) /\ Roots q by B9,C2;
         hence contradiction by B7,ZZ1d;
         end; then
       not b in {a} by TARSKI:def 1;
       hence o in S by C2,C3,B8,XBOOLE_0:def 3;
     end;
   hence thesis by C1,TARSKI:2;
   end; then
q is non zero with_roots & p is Ppoly of F,({a}\/S);
hence thesis by A,B3,C,ZZ1a;
end;
