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Rh By Add., Syll., $$\scriptstyle{\vdash}$$(1), The right side of (2) implies, by Syll., By Id., Perm., Add.$$\scriptstyle{\frac{p/p|p,\quad q}{\quad p,\qquad q}}$$, By Syll. twice, $$\scriptstyle{\vdash}$$(2), $$\scriptstyle{\vdash}$$(3), $$\scriptstyle{\vdash}$$(4),

(b) By lemma to Syll., $$\scriptstyle{q/q}$$; by Perm. and Syll., $$\scriptstyle{q/q}$$. Hence, $$\scriptstyle{q/q|L/L}$$; by Perm., $$\scriptstyle{L/L|q/q}$$.

Now, by Syll.: $\scriptstyle{L/L By $$\scriptstyle{\vdash}$$b, $$\scriptstyle{\vdash}$$a, and Taut.$$\scriptstyle{\frac{L}{p}}$$, result. We can now complete the proof of 'Association.'

Dem: By Syll., $$\scriptstyle{p}$$$$\scriptstyle{.\supset:q/r|r.}$$ By Syll. twice, $$\scriptstyle{\vdash}$$ Lemma, result.

Dem.: By Syll., Assoc., By (1)&emsp;$$\scriptstyle{\frac{s/s,\quad q,\quad p/p}{s,\quad\;\;r,\quad p\,}}$$, result.

Theorems Equivalent to the Definitions of $$\scriptstyle{p\supset q, p.q}$$, in Principia.

$$\scriptstyle{p\supset q.\supset.\sim p\vee q}$$, and reciprocal theorem.

Dem.: Taut., and Syll.

Reciprocal theorem by Add.$$\scriptstyle{\frac{s/s,\quad p}{\;s,\;\quad p}}$$, and Syll.

$$\scriptstyle{p|q.\supset.\sim p\vee\sim q}$$, and reciprocal theorem.