
theorem u2:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E 
for x,y being Element of FAdj(F,{a})
for p,q being Polynomial of F st x = Ext_eval(p,a) & y = Ext_eval(q,a)
holds x + y = Ext_eval(p+q,a) & x * y = Ext_eval(p*'q,a)
proof
let F be Field, E be FieldExtension of F; let a be F-algebraic Element of E; 
let x,y be Element of FAdj(F,{a}); let p,q be Polynomial of F;
assume AS: x = Ext_eval(p,a) & y = Ext_eval(q,a); 
H0: F is Subring of E by FIELD_4:def 1;
x in the carrier of FAdj(F,{a}); then
H1: x in carrierFA({a}) by FIELD_6:def 6;
y in the carrier of FAdj(F,{a}); then
y in carrierFA({a}) by FIELD_6:def 6; then
H2: [x,y] in [:carrierFA({a}),carrierFA({a}):] by H1,ZFMISC_1:def 2;
thus x + y = ((the addF of E)||carrierFA({a})).(x,y) by FIELD_6:def 6
          .= Ext_eval(p,a) + Ext_eval(q,a) by AS,H2,FUNCT_1:49
          .= Ext_eval(p+q,a) by H0,ALGNUM_1:15;
thus x * y = ((the multF of E)||carrierFA({a})).(x,y) by FIELD_6:def 6
          .= Ext_eval(p,a) * Ext_eval(q,a) by AS,H2,FUNCT_1:49
          .= Ext_eval(p*'q,a) by H0,ALGNUM_1:20;
end;
