reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;

theorem Th3:
  for x being bound_QC-variable of A, a being free_QC-variable of A,
  l, ll being FinSequence of QC-variables(A)
  holds
  f = a .--> x & ll = Subst(l,f) & 1 <= k & k <= len l implies (l.
  k = a implies ll.k = x) & (l.k <> a implies ll.k = l.k)
proof
  let x be bound_QC-variable of A, a be free_QC-variable of A,
  l, ll be FinSequence of QC-variables(A);
  set f9 = a .--> x;
  assume
A1: f = a .--> x & ll = Subst(l,f) & 1 <= k & k <= len l;
  thus l.k = a implies ll.k = x
  proof
A2: f9.a = x by FUNCOP_1:72;
    assume
A3: l.k = a;
    then l.k in { a } by TARSKI:def 1;
    then l.k in dom(f9);
    hence thesis by A1,A3,A2,Def1;
  end;
  assume l.k <> a;
  then not l.k in { a } by TARSKI:def 1;
  then not l.k in dom(f9);
  hence thesis by A1,Def1;
end;
