\magnification \magstep2
\nopagenumbers

%%%% Corollary 3 (2.1) in assoc4.pdf, establishing the alternating
%%%% 4-form (Bruck and Kleinfeld 1951).

$$
(x,yz,t) - (x,z,t) y - z (x,y,t) = (x,z,[y,t]) + ([x,z],y,t)
$$

\bye

%%%%%%%%

%% html tag

%%%% <img src="images/assoc4.png"
%%%%      alt="(x,yz,t) - (x,z,t) y - z (x,y,t) = (x,z,[y,t]) + ([x,z],y,t)"
%%%%      width="896" height="36">

%%%%%%%%

%% png creation (normal size, transparent)

%%% pdftex assoc4.tex
%%% pdftocairo -png -singlefile -transp -r 180 assoc4.pdf tmp
%%% magick -define png:exclude-chunk=date,time tmp.png -trim +repage assoc4.png
%%% optipng assoc4.png %% no effect

%%%% magick adds white (fallback) background to the transparent image
%%%% and has the convenient option -trim.

%%%% Image size is 896x36.

%% program versions

%%% This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdftex 2024.3.16)

%%% pdftocairo version 0.86.1
%%% ImageMagick 7.1.1-43 Q16-HDRI x86_64 22550 https://imagemagick.org
%%% OptiPNG version 0.7.7

%%%%%%%%

%%%% First version (using tex, dvipng):
%%% 2024-05-19

%%%% Recompiled:
%%% 2025-08-09

%%%%%%%%
