Section IntermediateField.
Variable M : {subfield L}.
Hypothesis (sKME : (K <= M <= E)%VS) (nKM : normalField K M).
Definition normalField_cast (x : gal_of E) : gal_of M := gal M x.

Lemma normalField_cast_eq x :
x \in 'Gal(E / K) -> {in M, normalField_cast x =1 x}.