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Rh Now, $$\scriptstyle{Q_1|\pi\mathsf{I}\pi}$$ fulfils (α) and (β). For (α) $$\scriptstyle{\pi}$$ being the complex expression $$\scriptstyle{t\mathsf{I}t|t}$$, is a case of the form $$\scriptstyle{q|r}$$, and (β) we have, by (c) above, $$\scriptstyle{\pi|\pi/Q_1}$$ and by (a) $$\scriptstyle{\pi\mathsf{I}\pi|Q_1}$$.

To obtain the strictest development of the proof we have only to write $$\scriptstyle{Q_1/\pi|\pi}$$ for $$\scriptstyle{P}$$ and $$\scriptstyle{\pi|\pi/Q_1}$$ for $$\scriptstyle{T}$$ all through the preceding argument. $$\scriptstyle{\left[\text{Gives }s\vee p.\supset.p\vee s\text{ by }\frac{p|p\quad s|s}{p\qquad s}.\right]}$$

Dem.: Prop. $$\scriptstyle{\frac{p\quad p\quad p}{p\quad q\quad r}}$$, Id., and Rule. Dem.: Id. $$\scriptstyle{\frac{p/p}{p}}$$, Perm., and Rule. $$\scriptstyle{\left[\text{Gives }s.\supset.p\vee s\text{ by }\frac{p/p}{p}.\right]}$$

Dem.: By Perm. (twice), $$\scriptstyle{s/s|p}$$&emsp;(a)

By Prop. $$\scriptstyle{\frac{\overline{p|s/s}\quad s/s\quad p\quad s}{\;\;p\qquad\;\,q\quad\;\;r\quad s},\;\vdash(a),\;\vdash}$$ Id. , $$\scriptstyle{s}$$

By Perm., result. Return from Generalised Implication $$\scriptstyle{P|\pi/Q}$$ to $$\scriptstyle{P|Q/Q}$$. Dem.: By Perm. (twice), $$\scriptstyle{p/s}$$&emsp;(a)

By Prop. $$\scriptstyle{\frac{\overline{s/p}\quad p\quad s\quad u}{\;p\quad\;\,q\quad r\quad s},\;\vdash}$$(a), $$\scriptstyle{u|p}$$ $$\scriptstyle{u}$$ Write $$\scriptstyle{p/p}$$ for $$\scriptstyle{u}$$: by Id. and Perm. (twice), result.