
theorem r58:
for R being domRing,
    S1,S2 being non empty finite Subset of R
for p being Ppoly of R,S1
for q being Ppoly of R,S2 st S1 /\ S2 = {} holds p *' q is Ppoly of R,(S1\/S2)
proof
let R be domRing, S1,S2 be non empty finite Subset of R;
let p be Ppoly of R,S1; let q be Ppoly of R,S2;
assume A: S1 /\ S2 = {};
reconsider B1 = Bag S1, B2 = Bag S2 as non zero bag of the carrier of R;
B: B1 = (S1,1)-bag & B2 = (S2,1)-bag by RING_5:def 1;
S1 misses S2 by A; then
B1 + B2 = (S1\/S2, 1)-bag by B,UPROOTS:10 .= Bag(S1 \/ S2) by RING_5:def 1;
hence thesis by RING_5:58;
end;
