Fixpoint PairwiseEqual (T : fol.System L) (n : nat) {struct n} :
 fol.Terms L n -> fol.Terms L n -> Prop :=
  match n return (fol.Terms L n -> fol.Terms L n -> Prop) with
  | O => fun ts ss : fol.Terms L 0 => True
  | S x =>
      fun ts ss : fol.Terms L (S x) =>
      let (a, b) := proj1_sig (consTerms L x ts) in
      let (c, d) := proj1_sig (consTerms L x ss) in
      SysPrf T (equal a c) /\ PairwiseEqual T x b d
  end.Remark subAllnVars1 :
 forall (a : nat) (ts ss : fol.Terms L a),
 ts = subAllTerms L _ (fst (nVars L a)) (termsMap a ts ss).Remark subAllnVars2 :
 forall (a : nat) (ts ss : fol.Terms L a),
 ss = subAllTerms L _ (snd (nVars L a)) (termsMap a ts ss).Remark addPairwiseEquals :
 forall (T : fol.System L) (n : nat) (ts ss : fol.Terms L n),
 PairwiseEqual T n ts ss ->
 forall m0 : nat -> fol.Term L,
 (forall q : nat, q < n + n -> m0 q = termsMap n ts ss q) ->
 forall f0 : fol.Formula L,
 SysPrf T
   (subAllFormula L
      (nat_rec (fun _ : nat => fol.Formula L) f0
         (fun (n : nat) (Hrecn : fol.Formula L) =>
          fol.impH L
            (fol.equal L (fol.var L (n + n)) (fol.var L (S (n + n)))) Hrecn)
         n) m0) -> SysPrf T (subAllFormula L f0 m0).

(* Specifications: *)
Lemma equalRelation : forall (T : fol.System L) (r : Relations L) (ts ss : fol.Terms L _), PairwiseEqual T _ ts ss -> SysPrf T (iffH (atomic r ts) (atomic r ss)).
Lemma equalFunction : forall (T : fol.System L) (f : Functions L) (ts ss : fol.Terms L _), PairwiseEqual T _ ts ss -> SysPrf T (equal (apply f ts) (apply f ss)).
Lemma subWithEqualsTerm : forall (a b t : fol.Term L) (v : nat) (T : fol.System L), SysPrf T (equal a b) -> SysPrf T (equal (substituteTerm L t v a) (substituteTerm L t v b)).
Lemma subWithEqualsTerms : forall (a b : fol.Term L) (n : nat) (ts : fol.Terms L n) (v : nat) (T : fol.System L), SysPrf T (equal a b) -> PairwiseEqual T _ (substituteTerms L _ ts v a) (substituteTerms L _ ts v b).
Lemma subWithEquals : forall (f : fol.Formula L) (v : nat) (a b : fol.Term L) (T : fol.System L), SysPrf T (equal a b) -> SysPrf T (impH (substituteFormula L f v a) (substituteFormula L f v b)).