Section TraceAndNormField.
Variables K E : {subfield L}.
End TraceAndNormField.
Definition normalField U V := [forall x in kAEndf U, x @: V == V]%VS.
Arguments normalFieldP {K E}.
Definition galois U V := [&& (U <= V)%VS, separable U V & normalField U V].

Lemma galoisS K M E : (K <= M <= E)%VS -> galois K E -> galois M E.