
theorem lemphi4:
for F being Field,
    E being FieldExtension of F
for T being non empty finite Subset of E
for x being T-evaluating Function of (card T),E
holds RAdj(F,T) = Image_hom_Ext_eval(x,F)
proof
let F be Field, E be FieldExtension of F, T be non empty finite Subset of E;
let x be T-evaluating Function of (card T),E;
set R = Image_hom_Ext_eval(x,F), S = RAdj(F,T), f = hom_Ext_eval(x,F);
set n = card T;
A: T is Subset of the carrier of R by lemphi4ab;
B: F is Subring of R by lemphi3;
   R is Subring of S & S is Subring of E by lemphi4aa; then
   R is Subring of E by lemring; then
C: RAdj(F,T) is Subring of Image_hom_Ext_eval(x,F) by A,B,FIELD_6:32;
   Image_hom_Ext_eval(x,F) is Subring of RAdj(F,T) by lemphi4aa;
hence thesis by C,FIELD_6:12;
end;
