
theorem extevalsubst:
for F being Field
for p,q being Polynomial of F
for E being FieldExtension of F
for a being Element of E
holds Ext_eval(Subst(p,q),a) = Ext_eval(p,Ext_eval(q,a))
proof
let F be Field, p,q be Polynomial of F; let E be FieldExtension of F;
let a be Element of E;
reconsider p1 = p, q1 = q as Polynomial of E by FIELD_4:8;
A: eval(Subst(p1,q1),a) = eval(p1,eval(q1,a)) by POLYNOM5:53;
      q is Element of the carrier of Polynom-Ring F &
      q1 is Element of the carrier of Polynom-Ring E
      by POLYNOM3:def 10; then
B: eval(q1,a) = Ext_eval(q,a) by FIELD_4:26;
      p is Element of the carrier of Polynom-Ring F &
      p1 is Element of the carrier of Polynom-Ring E
      by POLYNOM3:def 10; then
C: eval(Subst(p1,q1),a) = Ext_eval(p,Ext_eval(q,a)) by B,A,FIELD_4:26;
D:    Subst(p,q) is Element of the carrier of Polynom-Ring F &
      Subst(p1,q1) is Element of the carrier of Polynom-Ring E
      by POLYNOM3:def 10;
   Subst(p1,q1) = Subst(p,q) by extsubst;
hence thesis by D,C,FIELD_4:26;
end;
