Let be a
basis
of . On one hand, we can complete this basis, according to
fact,
to a basis of , on the other hand, we can complete it to a basis of . Then
-
is a
generating system
of . We claim that it is even a basis. To see this, let
-
This implies that the element
-
belongs to . From this we get directly
for
and
for
.
From this we can infer that also
for all . Hence, we have
linear independence.
This gives altogether