 This is a theorem of logic containing the free variables \(x\), \(y\), and \(F\). First, we instantiate \(x\) and \(y\) to \(a\) and \(b\), respectively. Then, we instantiate \(F\) to the concept \([\lambda z \, R^{+}(a,z) \amp Pz]\) and apply \(\lambda\)-Conversion (though Frege could simply use his Rule of Substitution to achieve the same inference). The concept being instantiated for \(F\) is the concept member of the R-series beginning with a and which falls under P. The result of instantiating the free variables in Fact (7) and then applying \(\lambda\)-Conversion yields a rather long conditional, with numerous conjuncts in the antecedent and the claim that \(Pb\) in the consequent. Thus, if the antecedent can be established, the proof is done. For those following along with pencil and paper, all of the conjuncts in the antecedent are things we already know, with the exception of the claim that \([\lambda z \, R^{+}(a,z)\amp Pz]\) is hereditary on \(R\). However, this claim can be established straightforwardly from things we know to be true (and, in particular, from facts contained in the antecedent of the Principle we are trying to prove, which we assumed as part of our conditional proof). The reader is encouraged to complete the proof as an exercise. For those who would like to check their work, we give the complete Proof of the General Principle of Induction here: 