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38 Dem.: Prop.$$\scriptstyle{\frac{Q|Q\quad\pi/Q\quad P}{\;p\qquad q,r\quad s},\;\vdash}$$ Lemma, result.

Hence, by Perm., $$\scriptstyle{P|Q/Q}$$, i.e. $$\scriptstyle{\left[\text{Gives }p\supset q.\supset:q\supset s.\supset.p\supset s\text{ for }\frac{q\quad s/s}{r\quad\;s\;}.\right]}$$

Dem.: In this Dem., Permutation is used to correct the twisting action of $$\scriptstyle{S'}$$, much as handwriting has first to be inverted, if it is to be seen right in a mirror.

By $$\scriptstyle{S'\frac{q/s\quad s/q\quad u}{\;p\quad\;\;q,r\quad s},\;\vdash}$$ Perm., and Perm., By $$\scriptstyle{S'\frac{s/q|u\quad u|s/q\quad\overline{q/s|u}}{\;p\qquad q,r\qquad s},\;\vdash}$$ Perm., $$\scriptstyle{\vdash\;a}$$, and Perm., By $$\scriptstyle{S'\frac{p|q/r\quad s/q\overline{|p/s}\quad\overline{q/s\overline{|p/s}}}{p\qquad\quad q,r\qquad\quad s},\;\vdash S',\;\vdash b}$$, result. The structure of the proof is this: $$\scriptstyle{\text{Syll.}\frac{p\quad q/r\quad r}{p\quad q,r\quad s}}$$ We now need only the Lemma $$\scriptstyle{q}$$ for our result to follow by Syll. twice. The proof of this lemma&mdash;call it L&mdash;is as follows: We prove (a) $$\scriptstyle{q|L/L}$$, (b) $$\scriptstyle{L/L|q/q}$$. From this, by Syll. and Tautol., the result follows.

Dem.: (a) By $$\scriptstyle{\text{Syll.}\frac{q,p}{r,s}}$$,