let c be Quaternion; :: thesis: c + 0q = c
A1: 0q = [*(In (0,REAL)),(In (0,REAL))*] by ARYTM_0:def 5
.= [*0,0,0,0*] by QUATERNI:91 ;
consider x, y, w, z being Element of REAL such that
A2: c = [*x,y,w,z*] by Lm1;
c + 0q = [*(x + 0),(y + 0),(w + 0),(z + 0)*] by A1, A2, QUATERNI:def 7
.= [*x,y,w,z*] ;
hence c + 0q = c by A2; :: thesis: verum