let c be Quaternion; :: thesis: c * 1q = c
A1: 1q = [*jj,(In (0,REAL))*] by ARYTM_0:def 5
.= [*1,0,0,0*] by QUATERNI:91 ;
consider x, y, w, z being Element of REAL such that
A2: c = [*x,y,w,z*] by Lm1;
c * 1q = [*((((x * 1) - (y * 0)) - (w * 0)) - (z * 0)),((((x * 0) + (y * 1)) + (w * 0)) - (z * 0)),((((x * 0) + (1 * w)) + (0 * z)) - (0 * y)),((((x * 0) + (z * jj)) + (y * 0)) - (w * 0))*] by A1, A2, QUATERNI:def 10
.= [*x,y,w,z*] ;
hence c * 1q = c by A2; :: thesis: verum