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Theorem,

$\scriptstyle {P|\pi /Q}$$\scriptstyle {Q/Q|P}$

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 {p|q/r}$$\scriptstyle {s/q}$$\scriptstyle {p/s}$ ($\scriptstyle {S'}$)
Syllogism,	$\scriptstyle {p|q/r}$$\scriptstyle {q/s}$$\scriptstyle {p/s}$

$\scriptstyle {\left[{\text{Gives }}p\supset q.\supset :q\supset s.\supset .p\supset s{\text{ for }}{\frac {q\quad s/s}{r\quad \;s\;}}.\right]}$

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.,

$\scriptstyle {q/s|u}$$\scriptstyle {u|s/q}$ (a)

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.,

$\scriptstyle {q/s|u}$$\scriptstyle {s/q|u}$ (b)

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.

Association,

$\scriptstyle {p}$$\scriptstyle {q/r}$$\scriptstyle {q}$$\scriptstyle {p/r}$

The structure of the proof is this:

$\scriptstyle {{\text{Syll.}}{\frac {p\quad q/r\quad r}{p\quad q,r\quad s}}}$

gives

$\scriptstyle {p}$$\scriptstyle {q/r}$$\scriptstyle {.\supset :q/r|r.}$$\scriptstyle {p/r}$.

We now need only the Lemma $\scriptstyle {q}$$\scriptstyle {q/r|r}$ for our result to follow by Syll. twice.

Lemma,

$\scriptstyle {q}$$\scriptstyle {q/p|p}$

The proof of this lemma—call it L—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 $\scriptstyle {{\text{Syll.}}{\frac {q,p}{r,s}}}$,

$\scriptstyle {p|q/q.\supset .q/p}$$\scriptstyle {p/p}$

(1)