let z be quaternion number ; :: thesis: ( z = 0 implies ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0 )
assume z = 0 ; :: thesis: ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0
then ( Rea z = 0 & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) by Lm5, Th23;
hence ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0 ; :: thesis: verum