A1: <j> * <k> = [*((((0 * 0 ) - (0 * 0 )) - (1 * 0 )) - (0 * 1)),((((0 * 0 ) + (0 * 0 )) + (1 * 1)) - (0 * 0 )),((((0 * 0 ) + (0 * 1)) + (0 * 0 )) - (1 * 0 )),((((0 * 1) + (0 * 0 )) + (0 * 0 )) - (1 * 0 ))*] by Def10
.= [*0 ,1,0 ,0 *] ;
<k> * <j> = [*((((0 * 0 ) - (0 * 0 )) - (0 * 1)) - (1 * 0 )),((((0 * 0 ) + (0 * 0 )) + (0 * 0 )) - (1 * 1)),((((0 * 1) + (0 * 0 )) + (0 * 1)) - (0 * 0 )),((((0 * 0 ) + (1 * 0 )) + (0 * 1)) - (0 * 0 ))*] by Def10
.= [*0 ,(- 1),0 ,0 *] ;
then (<j> * <k> ) + (<k> * <j> ) = [*(0 + 0 ),(0 + 0 ),(0 + 0 ),(1 + (- 1))*] by A1, Def7
.= 0 by Lm5 ;
hence <j> * <k> = - (<k> * <j> ) by Def8; :: thesis: verum