
theorem r59:
for F being Field,
    S1,S2 being non empty finite Subset of F
for p being Ppoly of F,S1
for a being Element of F, q being non constant Polynomial of F
st p = rpoly(1,a) *' q & S2 = S1 \ {a} holds q is Ppoly of F,S2
proof
let R be Field, S1,S2 be non empty finite Subset of R;
let p be Ppoly of R,S1; let a be Element of R,
    q be non constant Polynomial of R;
assume A: p = rpoly(1,a) *' q & S2 = S1 \ {a}; then
H: Roots rpoly(1,a) c= Roots p by ZZ3b,RING_4:1;
Roots rpoly(1,a) = {a} by RING_5:18; then
B: {a} c= S1 by H,RING_5:63;
C: now assume a in S2;
   then not a in {a} by A,XBOOLE_0:def 5;
   hence contradiction by TARSKI:def 1;
   end;
S2 \/ {a} = S1 \/ {a} by A,XBOOLE_1:39 .= S1 by B,XBOOLE_1:12;
hence thesis by A,C,r59a;
end;
