
theorem finex:
for F being Field,
    E being FieldExtension of F
for f being FinSequence of E st
for i being Nat st i in dom f holds f.i in F
holds f is FinSequence of F & Sum f in F
proof
let F be Field, E be FieldExtension of F, f be FinSequence of E;
assume AS: for i being Nat st i in dom f holds f.i in F;
I1: F is Subring of E by FIELD_4:def 1;
per cases;
suppose len f = 0;
  then A: f = <*>(the carrier of F);
  then Sum f = Sum <*>(the carrier of F) by I1,FIELD_4:2
            .= 0.F by RLVECT_1:43;
  hence f is FinSequence of F & Sum f in F by A;
  end;
suppose len f > 0;
  then f <> {};
  then consider G being FinSequence, y being object such that
  B2: f = G^<*y*> by FINSEQ_1:46;
  rng G c= rng f by B2,FINSEQ_1:29; then
  reconsider G as FinSequence of E by XBOOLE_1:1,FINSEQ_1:def 4;
  C: dom G c= dom f by B2,FINSEQ_1:26;
  D: now let i be Nat;
     assume C0: i in dom G;
     then G.i = f.i by B2,FINSEQ_1:def 7;
     hence G.i in F by C,C0,AS;
     end;
  now let o be object;
    assume o in rng G; then
    consider u being object such that
    D1: u in dom G & G.u = o by FUNCT_1:def 3;
    reconsider u as Element of NAT by D1;
    o in F by D,D1;
    hence o in the carrier of F;
    end;
  then rng G c= the carrier of F;
  then reconsider G as FinSequence of F by FINSEQ_1:def 4;
  rng<*y*> = {y} by FINSEQ_1:39;
  then G5: y in rng<*y*> by TARSKI:def 1;
  rng<*y*> c= rng f by B2,FINSEQ_1:30;
  then consider u being object such that
  G6: u in dom f & f.u = y by G5,FUNCT_1:def 3;
  reconsider u as Element of NAT by G6;
  f.u in F by AS,G6; then
  reconsider y as Element of F by G6;
  B4: <*y*> is FinSequence of F & G is FinSequence of F & f=G^<*y*> by B2;
  Sum f = Sum(G^<*y*>) by I1,FIELD_4:2,B2
       .= Sum G + Sum <*y*> by RLVECT_1:41
       .= Sum G + y by RLVECT_1:44;
  hence f is FinSequence of F & Sum f in F by B4;
  end;
end;
