
theorem sp:
for F being Field,
    E being FieldExtension of F
for K being E-extending FieldExtension of F
holds VecSp(E,F) is Subspace of VecSp(K,F)
proof
let F be Field, E be FieldExtension of F;
let K be E-extending FieldExtension of F;
set VK = VecSp(K,F), VE = VecSp(E,F);
H1: E is Subring of K by FIELD_4:def 1; then
H2: the carrier of E c= the carrier of K by C0SP1:def 3;
H3: the carrier of VE = the carrier of E &
    the carrier of VK = the carrier of K by FIELD_4:def 6;
F is Subring of E by FIELD_4:def 1; then
H4: the carrier of F c= the carrier of E by C0SP1:def 3;
B1: the carrier of VE c= the carrier of VK by H1,H3,C0SP1:def 3;
B2: 0.VE = 0.E by FIELD_4:def 6
        .= 0.K by H1,C0SP1:def 3
        .= 0.VK by FIELD_4:def 6;
B3: the addF of VE = the addF of E by FIELD_4:def 6
                  .= (the addF of K)||the carrier of E by H1,C0SP1:def 3
                  .= (the addF of VK)||the carrier of E by FIELD_4:def 6
                  .= (the addF of VK)||the carrier of VE by FIELD_4:def 6;
B4: the lmult of VE
        = (the multF of E)|[:the carrier of F,the carrier of VE:]
          by H3,FIELD_4:def 6
       .= ((the multF of K)||the carrier of E) |
                [:the carrier of F,the carrier of VE:] by H1,C0SP1:def 3
       .= (the multF of K) | [:the carrier of F,the carrier of VE:]
          by H4,H3,ZFMISC_1:96,RELAT_1:74
       .= ((the multF of K)|[:the carrier of F,the carrier of K:])
                               | [:the carrier of F, the carrier of VE:]
          by H2,H3,ZFMISC_1:96,RELAT_1:74
       .= (the lmult of VK) | [:the carrier of F, the carrier of VE:]
          by FIELD_4:def 6;
thus thesis by B1,B2,B3,B4,VECTSP_4:def 2;
end;
