
theorem quah2:
for F being Field
for E being FieldExtension of F
holds deg(E,F) = 1 iff {1.E} is Basis of VecSp(E,F)
proof
let F be Field; let E be FieldExtension of F;
set V = VecSp(E,F);
H0: F is Subring of E & F is Subfield of E by FIELD_4:def 1,FIELD_4:7;
reconsider e = 1.E as Element of V by FIELD_4:def 6;
Z: now assume deg(E,F) = 1; then
   AS1: the carrier of E = the carrier of F by quah1;
reconsider A = {e} as Subset of V;
0.V = 0.E by FIELD_4:def 6;
then L: A is linearly-independent by VECTSP_7:3;
H1: the carrier of Lin(A) =
the set of all Sum(l) where l is Linear_Combination of A by VECTSP_7:def 2;

H2: now let o be object;
    assume o in the carrier of V;
    then reconsider v = o as Element of the carrier of V;
    reconsider a = v as Element of E by FIELD_4:def 6;
    defpred P[object,object] means ($1 = e & $2 = a) or ($1 <> e & $2 = 0.E);
    G0: for x being object st x in the carrier of V
        ex y being object st y in the carrier of E & P[x,y]
        proof
        let o be object;
        assume o in the carrier of V;
        per cases;
        suppose A: o = e; take a; thus thesis by A; end;
        suppose A: o <> e; take 0.E; thus thesis by A; end;
        end;
    consider f being Function of the carrier of V, the carrier of E such that
    G1: for x being object st x in the carrier of V holds P[x,f.x]
        from FUNCT_2:sch 1(G0);
    dom f = the carrier of V & rng f c= the carrier of E by FUNCT_2:def 1; then
    reconsider f as Element of Funcs(the carrier of V, the carrier of F)
       by AS1,FUNCT_2:def 2;
    ex T being finite Subset of V st
    for v being Element of V st not v in T holds f.v = 0.F
      proof
      reconsider T = {e} as finite Subset of V;
      take T;
      now let u be Element of V;
        assume not u in T;
        then u <> e by TARSKI:def 1;
        hence f.u = 0.E by G1 .= 0.F by H0,C0SP1:def 3;
        end;
      hence thesis;
      end;
    then reconsider l = f as Linear_Combination of V by VECTSP_6:def 1;
    now let o be object;
      assume o in Carrier l; then
      o in {v where v is Element of V: l.v <> 0.F} by VECTSP_6:def 2;
      then consider u being Element of V such that
      I: o = u & l.u <> 0.F;
      l.u <> 0.E by I,H0,C0SP1:def 3; then
      u = e by G1;
      hence o in A by I,TARSKI:def 1;
      end;
    then Carrier l c= A;
    then reconsider l as Linear_Combination of A by VECTSP_6:def 4;
    Sum l = l.e * e by VECTSP_6:17
         .= ((the multF of E)|[:the carrier of F,the carrier of E:]).(l.e,e)
            by FIELD_4:def 6
         .= a * 1.E by AS1,G1
         .= v;
    hence o in the set of all Sum(l) where l is Linear_Combination of A;
    end;

now let o be object;
  assume o in the set of all Sum(l) where l is Linear_Combination of A;
  then consider l being Linear_Combination of A such that G: o = Sum l;
  thus o in the carrier of V by G;
  end;
then the carrier of V = the set of all Sum(l) where
                              l is Linear_Combination of A by H2,TARSKI:2;
   hence {1.E} is Basis of VecSp(E,F) by H1,L,VECTSP_4:31,VECTSP_7:def 3;

   end;
now assume M: {1.E} is Basis of VecSp(E,F);
  H4: card {1.E} = 1 by CARD_1:30;
  H5: V is finite-dimensional by M,MATRLIN:def 1; then
  dim V = 1 by M,H4,VECTSP_9:def 1;
  hence deg(E,F) = 1 by H5,FIELD_4:def 7;
  end;
hence thesis by Z;
end;
