End ALMOST_RING.
Variable Rth : ring_theory 0 1 radd rmul rsub ropp req.
Let ARth := Rth_ARth Rsth Reqe Rth.
Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext4.

Lemma gen_phiZ1_pos_sub : forall x y,
gen_phiZ1 (Z.pos_sub x y) == gen_phiPOS1 x + -gen_phiPOS1 y.