number-theory

150 9 Technical detailsThe first step towards precision is to clarify the relationship between rp and F.If P is not in the body of a A-abstract in M then clearly(2) rpgr.But now suppose P is in the bodies of some A-abstracts in M; say there are ndistinct such abstracts:1x,Nt, ... , -1x :Nn,with P in N; for i = 1, ..., n. By 9A5.1 these components must be linearly orderedby the relation "is in"; by renaming them if necessary, we can assume thatP is in Ni in 2x1Nl in N2 in 2x2N2 ... in Nn in Ax --N...By the subject-construction theorem (2B2) i must contain n applications of (-1)below (1), and each of these has one of the following forms, depending on whetherxi E FV(Ni) or not:(3)Fi -(Here 1 < i< n.) Hence(4) Fp F U {xlNi:giri HFi H Ni:q,If M has no bound-variable clashes then xl,... , xn are distinct and the right-handside of (4) is consistent. (But if M has bound-variable clashes an might beinconsistent with an (j * i) or even with a member of F.)The replacement lemmas will now be stated. The first one is easy and is used inproving the subject-reduction theorem (2C1). The second is more complex to statebut gives a stronger result; it is applied in 5B2.1(ii) which is used in 8F2.9C5 First Replacement Lemma for Deductions Let 0 be a TAA-deduction of theformula r F-> M:T and let M contain a component P with a position p. Then Acontains a deduction Op of a formula with formrp ' ->P :apfor some rp and vp; let {T/P}PM be the result of replacing P at p by a term T suchthatfor some FT c rp. ThenProof Let r;.deduction AT ofF 1-1FT FA T :Qp{T/P}pM:i.FT P T. Then Subjects(FT) = FV(T) and by 2A11(i) there is aFT H T :up.Replace Ap by AT in A and modify the contexts and subjects in all the formulae