let c be quaternion number ; :: thesis: c + 0q = c
A1: 0q = [*0,0*] by ARYTM_0:def 7
.= [*0,0,0,0*] by Lm4 ;
c in QUATERNION by Def2;
then consider x, y, w, z being Element of REAL such that
A2: c = [*x,y,w,z*] by Th7;
c + 0q = [*(x + 0),(y + 0),(w + 0),(z + 0)*] by A1, A2, Def7
.= [*x,y,w,z*] ;
hence c + 0q = c by A2; :: thesis: verum