Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/141

SECTION A] $$\scriptstyle{\vdash:.p\supset q.\supset:p.r.\supset.q.r}$$

This principle shows that we may multiply both sides of an implication by a common factor; hence it is called by Peano the "principle of the factor." We shall refer to it as "Fact." It is the analogue, for multiplication, of the primitive proposition *1·6.

Dem.

$$\scriptstyle{\vdash:.p\supset r.q\supset s.\supset:p.q.\supset.r.s}$$

This proposition, or rather its analogue for classes, was proved by Leibniz, and evidently pleased him, since he calls it "præclarum theorema ."

Dem.

$$\scriptstyle{\vdash:.p\supset r.q\supset s.\supset:p\or q.\supset.r\or s}$$

This theorem is the analogue of *3·47.

Dem.