let z1 be quaternion number ; :: thesis: for x being Real st z1 = x holds
( Rea (z1 * <j> ) = 0 & Im1 (z1 * <j> ) = 0 & Im2 (z1 * <j> ) = x & Im3 (z1 * <j> ) = 0 )
let x be Real; :: thesis: ( z1 = x implies ( Rea (z1 * <j> ) = 0 & Im1 (z1 * <j> ) = 0 & Im2 (z1 * <j> ) = x & Im3 (z1 * <j> ) = 0 ) )
assume A1: z1 = x ; :: thesis: ( Rea (z1 * <j> ) = 0 & Im1 (z1 * <j> ) = 0 & Im2 (z1 * <j> ) = x & Im3 (z1 * <j> ) = 0 )
( Rea (z1 * <j> ) = ((((Rea z1) * (Rea <j> )) - ((Im1 z1) * (Im1 <j> ))) - ((Im2 z1) * (Im2 <j> ))) - ((Im3 z1) * (Im3 <j> )) & Im1 (z1 * <j> ) = ((((Rea z1) * (Im1 <j> )) + ((Im1 z1) * (Rea <j> ))) + ((Im2 z1) * (Im3 <j> ))) - ((Im3 z1) * (Im2 <j> )) & Im2 (z1 * <j> ) = ((((Rea z1) * (Im2 <j> )) + ((Im2 z1) * (Rea <j> ))) + ((Im3 z1) * (Im1 <j> ))) - ((Im1 z1) * (Im3 <j> )) & Im3 (z1 * <j> ) = ((((Rea z1) * (Im3 <j> )) + ((Im3 z1) * (Rea <j> ))) + ((Im1 z1) * (Im2 <j> ))) - ((Im2 z1) * (Im1 <j> )) ) by Lm16;
hence ( Rea (z1 * <j> ) = 0 & Im1 (z1 * <j> ) = 0 & Im2 (z1 * <j> ) = x & Im3 (z1 * <j> ) = 0 ) by A1, Lm17, Th31; :: thesis: verum