End AlmostField.
Section Field.
Variable Fth : field_theory rO rI radd rmul rsub ropp rdiv rinv req.
Let Rth := (F_R Fth).
Let rI_neq_rO := (F_1_neq_0 Fth).
Let rdiv_def := (Fdiv_def Fth).
Let rinv_l := (Finv_l Fth).
Let AFth := F2AF Rsth Reqe Fth.
Let ARth := Rth_ARth Rsth Reqe Rth.

Lemma ring_S_inj x y : 1+x==1+y -> x==y.
Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0.