Definition checkPrfEQ4 (p recs : nat) :=
let r := cPairPi2 (cPairPi2 p) in
let A := cPair (S (S (S (S r)))) (codeNVars1 (pred (codeArityR r))) in
let B := cPair (S (S (S (S r)))) (codeNVars2 (pred (codeArityR r))) in
notZero (codeArityR r) *
charFunction 2 beq_nat (codeAxmEqHelp (pred (codeArityR r)) (codeIff A B))
(cPairPi1 p).

Lemma checkPrfEQ4IsPR : isPR 2 checkPrfEQ4.
Definition checkPrfEQ5 (p recs : nat) :=
let f := cPairPi2 (cPairPi2 p) in
notZero (codeArityF f) *
charFunction 2 beq_nat
(codeAxmEqHelp (pred (codeArityF f))
(cPair 0
(cPair (cPair (S f) (codeNVars1 (pred (codeArityF f))))
(cPair (S f) (codeNVars2 (pred (codeArityF f)))))))
(cPairPi1 p).