
theorem ZZ1d:
for F being Field
for p,q being non zero Polynomial of F st
for a being Element of F st a is_a_root_of p*'q holds multiplicity(p*'q,a) = 1
holds Roots p /\ Roots q = {}
proof
let F be Field, p,q be non zero Polynomial of F;
assume AS:
for a being Element of F st a is_a_root_of p*'q holds multiplicity(p*'q,a) = 1;
set a = the Element of Roots p /\ Roots q;
now assume Roots p /\ Roots q <> {}; then
  B: a in Roots p & a in Roots q by XBOOLE_0:def 4; then
  reconsider a as Element of F;
  D: a is_a_root_of p & a is_a_root_of q by B,POLYNOM5:def 10; then
  C: a is_a_root_of p*'q &
     multiplicity(p,a) >= 1 & multiplicity(q,a) >= 1 by ro1,UPROOTS:52;
  multiplicity(p*'q,a) = multiplicity(p,a) + multiplicity(q,a) by UPROOTS:55;
  then multiplicity(p*'q,a) >= 1 + 1 by C,XREAL_1:7;
  then multiplicity(p*'q,a) > 1 by NAT_1:13;
  hence contradiction by AS,D,ro1;
  end;
hence thesis;
end;
