
theorem mpa1:
for F being 0-characteristic Field,
    p being irreducible Element of the carrier of Polynom-Ring F
for E being FieldExtension of F
for a being Element of E st a is_a_root_of p,E holds multiplicity(p,a) = 1
proof
let F be 0-characteristic Field,
    p be irreducible Element of the carrier of Polynom-Ring F;
let E being FieldExtension of F, a be Element of E;
assume AS: a is_a_root_of p,E;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
   by FIELD_4:10; then
reconsider p1 = p as Element of the carrier of Polynom-Ring E;
   p <> 0_.(F); then
   p1 <> 0_.(E) by FIELD_4:12; then
reconsider p1 as non zero Element of the carrier of Polynom-Ring E
   by UPROOTS:def 5;
reconsider a1 = a as Element of E;
A: 0.E = Ext_eval(p,a) by AS,FIELD_4:def 2 .= eval(p1,a1) by FIELD_4:26;
B: (Deriv E).p1 = (Deriv F).p by mm5;
p gcd (Deriv F).p = 1_.F by mm6; then
p1 gcd (Deriv E).p1 = 1_.F by B,lemgcd .= 1_.E by FIELD_4:14; then
C: p1 is square-free by mm4i;
D: now assume multiplicity(p1,a1) > 1; then
   E: multiplicity(p1,a1) >= 1 + 1 by INT_1:7;
   (X-a1)`^multiplicity(p1,a1) divides p1 by multi2; then
   (X-a1)`^2 divides p1 by E,multi1;
   hence contradiction by C,lemsq;
   end;
multiplicity(p1,a1) >= 1 by A,POLYNOM5:def 7,UPROOTS:52;
hence thesis by D,XXREAL_0:1,defM;
end;
