
theorem BE1:
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for BE being non empty linearly-independent Subset of VecSp(E,F),
    BK being non empty linearly-independent Subset of VecSp(K,E)
holds card Base(BE,BK) = card [:BE,BK:]
proof
let F be Field, E be FieldExtension of F, K be F-extending FieldExtension of E;
let BE be non empty linearly-independent Subset of VecSp(E,F),
    BK be non empty linearly-independent Subset of VecSp(K,E);
defpred P[object,object] means
  ex a,b being Element of K st a in BE & b in BK & $1 = a * b & $2 = [a,b];
A: now let x be object;
   assume x in Base(BE,BK); then
   consider a,b being Element of K such that
   B1: x = a * b & a in BE & b in BK;
   thus ex y being object st y in [:BE,BK:] & P[x,y]
     proof
     take [a,b];
     thus thesis by B1,ZFMISC_1:def 2;
     end;
   end;
consider f being Function of Base(BE,BK),[:BE,BK:] such that
B: for x being object st x in Base(BE,BK)  holds P[x,f.x]
   from FUNCT_2:sch 1(A);
C: dom f = Base(BE,BK) by FUNCT_2:def 1;
H1: the carrier of VecSp(K,E) = the carrier of K &
    the carrier of VecSp(E,F) = the carrier of E by FIELD_4:def 6;
H2: 0.VecSp(K,E) = 0.K by FIELD_4:def 6;
E is Subring of K by FIELD_4:def 1; then
H5: the carrier of E c= the carrier of K by C0SP1:def 3;
D: rng f = [:BE,BK:]
   proof
   D1: now let o be object;
       assume o in [:BE,BK:]; then
       consider a,b being object such that
       E1: a in BE & b in BK & o = [a,b] by ZFMISC_1:def 2;
       reconsider a,b as Element of K by E1,H1,H5;
       E0: a * b in Base(BE,BK) by E1;
       a * b in dom f by E1,C;
       then E2: f.(a*b) in rng f by FUNCT_1:3;
       consider a1,b1 being Element of K such that
       E3: a1 in BE & b1 in BK & a * b = a1 * b1 & f.(a*b) = [a1,b1] by E0,B;
       E4: b1 = b by BE0,E3,E1;
       reconsider bV = b as Element of VecSp(K,E) by FIELD_4:def 6;
       {b} c= BK by E1,TARSKI:def 1; then
       {bV} is linearly-independent by VECTSP_7:1; then
       E5: b <> 0.K by H2,VECTSP_7:3;
       b * a = a1 * b by E3,E4,GROUP_1:def 12
            .= b * a1 by GROUP_1:def 12;
       hence o in rng f by E1,E2,E3,E4,E5,VECTSP_1:5;
       end;
   rng f c= [:BE,BK:];
   hence thesis by D1,TARSKI:2;
   end;
f is one-to-one
  proof
  now let x1,x2 be object;
    assume F0: x1 in Base(BE,BK) & x2 in Base(BE,BK) & f.x1 = f.x2;
    then consider a1,b1 being Element of K such that
    F1: x1 = a1 * b1 & a1 in BE & b1 in BK;
    consider a2,b2 being Element of K such that
    F2: x2 = a2 * b2 & a2 in BE & b2 in BK by F0;
    F3: f.(a1*b1) = [a1,b1]
        proof
        P[a1*b1,f.(a1*b1)] by F0,F1,B; then
        consider a3,b3 being Element of K such that
        E3: a3 in BE & b3 in BK & a1 * b1 = a3 * b3 & f.(a3*b3) = [a3,b3];
        E4: b3 = b1 by BE0,F1,E3;
        reconsider b1V = b1 as Element of VecSp(K,E) by FIELD_4:def 6;
        {b1} c= BK by F1,TARSKI:def 1; then
        {b1V} is linearly-independent by VECTSP_7:1; then
        E5: b1 <> 0.K by H2,VECTSP_7:3;
        b1 * a1 = a3 * b1 by E3,E4,GROUP_1:def 12
               .= b1 * a3 by GROUP_1:def 12;
        hence thesis by E4,E3,E5,VECTSP_1:5;
        end;
    f.(a2*b2) = [a2,b2]
        proof
        P[a2*b2,f.(a2*b2)] by F0,F2,B; then
        consider a3,b3 being Element of K such that
        E3: a3 in BE & b3 in BK & a2 * b2 = a3 * b3 & f.(a3*b3) = [a3,b3];
        E4: b3 = b2 by BE0,F2,E3;
        reconsider b2V = b2 as Element of VecSp(K,E) by FIELD_4:def 6;
        {b2} c= BK by F2,TARSKI:def 1; then
        {b2V} is linearly-independent by VECTSP_7:1; then
        E5: b2 <> 0.K by H2,VECTSP_7:3;
        b2 * a2 = a3 * b2 by E3,E4,GROUP_1:def 12
               .= b2 * a3 by GROUP_1:def 12;
        hence thesis by E4,E3,E5,VECTSP_1:5;
        end;
    hence x1 = x2 by F0,F1,F2,F3;
    end;
  hence thesis by FUNCT_2:19;
  end;
hence thesis by C,D,CARD_1:70;
end;
