A1: <i> = [*0 ,1,0 ,0 *] by Def6', Lmx;
then A2: <i> * <j> = [*((((0 * 0 ) - (1 * 0 )) - (0 * 1)) - (0 * 0 )),((((0 * 0 ) + (1 * 0 )) + (0 * 0 )) - (0 * 1)),((((0 * 1) + (0 * 0 )) + (0 * 0 )) - (0 * 1)),((((0 * 0 ) + (0 * 0 )) + (1 * 1)) - (0 * 0 ))*] by Def10
.= [*0 ,0 ,0 ,1*] ;
<j> * <i> = [*((((0 * 0 ) - (0 * 1)) - (1 * 0 )) - (0 * 0 )),((((0 * 1) + (0 * 0 )) + (1 * 0 )) - (0 * 0 )),((((0 * 0 ) + (0 * 1)) + (1 * 0 )) - (0 * 0 )),((((0 * 0 ) + (0 * 0 )) + (0 * 0 )) - (1 * 1))*] by A1, Def10
.= [*0 ,0 ,0 ,(- 1)*] ;
then (<i> * <j> ) + (<j> * <i> ) = [*(0 + 0 ),(0 + 0 ),(0 + 0 ),(1 + (- 1))*] by A2, Def7
.= 0 by Lm5 ;
hence <i> * <j> = - (<j> * <i> ) by Def8; :: thesis: verum