let z be quaternion number ; :: thesis: for x being Real st z = x holds
( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
let x be Real; :: thesis: ( z = x implies ( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) )
assume A1: z = x ; :: thesis: ( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
A2: x = [*x,0 *] by ARYTM_0:def 7;
[*x,0 *] = [*x,0 ,0 ,0 *] by Lm4;
hence ( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) by A1, A2, Th23; :: thesis: verum