
theorem multiiso:
for F being Field
for p being non zero Polynomial of F
for E1,E2 being FieldExtension of F
for i being Function of E1,E2 st i is F-fixing isomorphism
for a being Element of E1 holds multiplicity(p,a) = multiplicity(p,i.a)
proof
let F be Field, p be non zero Polynomial of F;
let E1,E2 be FieldExtension of F; let i be Function of E1,E2;
assume AS: i is F-fixing isomorphism;
let a be Element of E1;
set n = multiplicity(p,a);
reconsider E2 as E1-isomorphic FieldExtension of F by AS,RING_3:def 4;
reconsider i as Isomorphism of E1,E2 by AS;
   p is Polynomial of E1 by FIELD_4:8; then
reconsider q = p as Element of the carrier of Polynom-Ring E1
   by POLYNOM3:def 10;
   p <> 0_.(F); then
   p <> 0_.(E1) by FIELD_4:12; then
reconsider q as non zero Element of the carrier of Polynom-Ring E1
   by UPROOTS:def 5;
reconsider E3 = E2 as E1-homomorphic Field;
reconsider h = i as additive Function of E1,E3;
H: now assume I1: (PolyHom h).q = 0_.(E2);
   I2: (PolyHom h).(0_.(E1)) = 0_.(E2) by FIELD_1:22;
   0_.(E1) in the carrier of Polynom-Ring E1 by POLYNOM3:def 10;
   hence contradiction by I1,I2,FUNCT_2:19;
   end; then
reconsider iq = (PolyHom h).q as non zero Polynomial of E2 by UPROOTS:def 5;
reconsider r = q as non zero Element of the carrier of Polynom-Ring E1;
reconsider hr = (PolyHom h).r as Polynomial of E3;
reconsider hr as non zero Polynomial of E3 by H,UPROOTS:def 5;
reconsider Xan = (X-a)`^n as Element of the carrier of Polynom-Ring E1
   by POLYNOM3:def 10;
reconsider Xan1 = (X-a)`^(n+1) as Element of the carrier of Polynom-Ring E1
   by POLYNOM3:def 10;

H: multiplicity(p,a) = multiplicity(q,a) by FIELD_14:def 6; then
A: (X-a)`^n divides q & not (X-a)`^(n+1) divides q by FIELD_14:67;
B: (PolyHom h).Xan divides (PolyHom h).r by H,FIELD_12:3,FIELD_14:67;
C: now assume C1: (PolyHom h).Xan1 divides (PolyHom h).r;
   reconsider E1 as E2-isomorphic FieldExtension of F by RING_3:74;
   reconsider k = i" as Isomorphism of E2,E1 by RING_3:73;
   reconsider E4 = E1 as E2-monomorphic E2-homomorphic Field;
   reconsider j = k as Monomorphism of E2,E4;
   reconsider iXan = (PolyHom h).Xan1, ir = (PolyHom h).r
                               as Element of the carrier of Polynom-Ring E2;
   C3: (PolyHom j).((PolyHom h).Xan1) = Xan1
       proof
       reconsider u = (PolyHom j).iXan as Polynomial of E1;
       now let o be object;
         assume o in NAT;
         then reconsider m = o as Nat;
         C4: dom h = the carrier of E1 by FUNCT_2:def 1;
         u.m = j.(iXan.m) by FIELD_1:def 2
            .= j.(h.(Xan1.m)) by FIELD_1:def 2 .= Xan1.m by C4,FUNCT_1:34;
         hence u.o = Xan1.o;
         end;
       hence thesis by FUNCT_2:12;
       end;
   (PolyHom j).((PolyHom h).r) = r
       proof
       reconsider u = (PolyHom j).ir as Polynomial of E1;
       now let o be object;
         assume o in NAT;
         then reconsider m = o as Nat;
         C4: dom h = the carrier of E1 by FUNCT_2:def 1;
         u.m = j.(ir.m) by FIELD_1:def 2
            .= j.(h.(r.m)) by FIELD_1:def 2 .= r.m by C4,FUNCT_1:34;
         hence u.o = r.o;
         end;
       hence thesis by FUNCT_2:12;
       end;
   hence contradiction by C3,C1,FIELD_12:3,A;
   end;
   (PolyHom h).Xan = (X-(h.a))`^n &
   (PolyHom h).Xan1 = (X-(h.a))`^(n+1) by aXn; then
D: multiplicity(hr,h.a) = n by B,C,FIELD_14:67;
   (PolyHom h).p = p
   proof
   reconsider u = (PolyHom h).q as Polynomial of E3;
   now let o be object;
     assume o in rng u; then
     consider v being object such that
     E: v in dom u & u.v = o by FUNCT_1:def 3;
     dom u = NAT by FUNCT_2:def 1; then
     reconsider v as Nat by E;
     u.v = h.(p.v) by FIELD_1:def 2 .= p.v by AS;
     hence o in the carrier of F by E;
     end; then
   rng u c= the carrier of F; then
   E: (PolyHom h).p is Function of NAT,the carrier of F by FUNCT_2:6;
   now let o be object;
     assume o in NAT;
     then reconsider m = o as Nat;
     u.m = h.(p.m) by FIELD_1:def 2 .= p.m by AS;
     hence u.o = p.o;
     end;
   hence thesis by E,FUNCT_2:12;
   end;
hence thesis by D,FIELD_14:def 6;
end;
