Section ALMOST_RING.
Variable ARth : almost_ring_theory 0 1 radd rmul rsub ropp req.
Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext3.
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.
End ZMORPHISM.
Proof (Eqsth N).
Proof (Eq_s_ext N.add N.mul).
Definition Nsub := SRsub N.add.
Definition Nopp := (@SRopp N).
Proof (SReqe_Reqe Nseqe).

Lemma Nath :
almost_ring_theory 0%N 1%N N.add N.mul Nsub Nopp (@eq N).