
theorem lembas1:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
for p being Polynomial of F holds Ext_eval(p,a) in Lin(Base a)
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E; let p be Polynomial of F;
set ma = MinPoly(a,F), r = p mod ma;
B: F is Subring of E by FIELD_4:def 1;
C: p = (p div ma) *' ma + r by RING_4:4;
D: deg r < deg ma by FIELD_5:16;
Ext_eval(p,a)
 = Ext_eval((p div ma) *' ma,a) + Ext_eval(r,a) by C,B,ALGNUM_1:15
.= (Ext_eval(p div ma,a) * Ext_eval(ma,a)) + Ext_eval(r,a) by B,ALGNUM_1:20
.= (Ext_eval(p div ma,a) * 0.E) + Ext_eval(r,a) by mpol2
.= Ext_eval(r,a);
hence thesis by D,lembas1a;
end;
