
theorem degmult:
for F being Field,
    E being F-finite FieldExtension of F,
    K being E-finite F-extending FieldExtension of E
holds deg(K,F) = deg(K,E) * deg(E,F)
proof
let F be Field, E be F-finite FieldExtension of F,
    K be E-finite F-extending FieldExtension of E;
set BE = the Basis of VecSp(E,F), BK = the Basis of VecSp(K,E);
BE is finite; then
B1: VecSp(E,F) is finite-dimensional by MATRLIN:def 1;
BK is finite; then
B2: VecSp(K,E) is finite-dimensional by MATRLIN:def 1;
A: Base(BE,BK) is Basis of VecSp((K qua FieldExtension of F),F) by BasKEF; then
B: VecSp(K,F) is finite-dimensional by MATRLIN:def 1;
hence deg(K,F) = dim VecSp(K,F) by FIELD_4:def 7
  .= card Base(BE,BK) by VECTSP_9:def 1,B,A
  .= card BE * card BK by BE2
  .= dim VecSp(E,F) * card BK by VECTSP_9:def 1,B1
  .= deg(E,F) * card BK by FIELD_4:def 7
  .= deg(E,F) * dim VecSp(K,E) by VECTSP_9:def 1,B2
  .= deg(K,E) * deg(E,F) by FIELD_4:def 7;
end;
