Section Substitution_Properties.
Variable L : Language.
Let Formula := Formula L.
Let Formulas := Formulas L.
Let System := System L.
Let Term := Term L.
Let Terms := Terms L.
Let var := var L.
Let apply := apply L.
Let equal := equal L.
Let atomic := atomic L.
Let impH := impH L.
Let notH := notH L.
Let forallH := forallH L.
Let orH := orH L.
Let andH := andH L.
Let existH := existH L.
Let iffH := iffH L.
Let SysPrf := SysPrf L.

Lemma freeVarSubTerms3 :
forall (n : nat) (ts : fol.Terms L n) (v : nat) (s : Term) (x : nat),
In x (freeVarTerms L n (substituteTerms L n ts v s)) ->
In x (list_remove _ eq_nat_dec v (freeVarTerms L n ts)) \/
In x (freeVarTerm L s).