PROPOSITION 25.

If from an even number an odd number be subtracted, the remainder will be odd.

For from the even number AB let the odd number BC be subtracted ;

I say that the remainder CA is odd.

A C D B

For let the unit CD be subtracted from BC ;

[VII. Def. 7]therefore DB is even.
But AB is also even ;
[IX. 24]therefore the remainder AD is also even.
And CD is an unit ;
[VII. Def. 7]therefore CA is odd.
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.6" [label="VII.def.6", fillcolor="#224422", tooltip="VII.def.6: An even number is that which is divisible into two equal parts.", URL="../../../book_vii/definitions/index.html#vii-def-6", target="_top"]; "VII.def.7" [label="VII.def.7", fillcolor="#224422", tooltip="VII.def.7: An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.", URL="../../../book_vii/definitions/index.html#vii-def-7", target="_top"]; "IX.24" [label="IX.24", fillcolor="#222244", tooltip="IX.24: If from an even number an even number be subtracted, the remainder will be even.", URL="../24/index.html", target="_top"]; "IX.25" [label="IX.25", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="IX.25: If from an even number an odd number be subtracted, the remainder will be odd.", URL="index.html", target="_top"]; "VII.def.7" -> "VII.def.1"; "VII.def.7" -> "VII.def.6"; "IX.24" -> "VII.def.6"; "IX.25" -> "VII.def.6"; "IX.25" -> "VII.def.7"; "IX.25" -> "IX.24"; }