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.
Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0.
Let gen_phiPOS_inject :=
gen_phiPOS_inj AFth ring_S_inj gen_phiPOS_not_0.

Lemma gen_phiZ_complete x y :
gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y ->
Zeq_bool x y = true.