let z be quaternion number ; :: thesis: ( z = 0 implies ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0 )
assume A1: z = 0 ; :: thesis: ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0
then A2: Rea z = 0 by Lm7, Th23;
A3: Im1 z = 0 by A1, Lm7, Th23;
Im2 z = 0 by A1, Lm7, Th23;
hence ((((Rea z) ^2 ) + ((Im1 z) ^2 )) + ((Im2 z) ^2 )) + ((Im3 z) ^2 ) = 0 by A1, A2, A3, Lm7, Th23; :: thesis: verum