reserve q,r,c,c1,c2,c3 for Quaternion;
reserve x1,x2,x3,x4,y1,y2,y3,y4 for Real;

theorem Th15:
  1q * c = c
proof
A1: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*1,0,0,0*] by QUATERNI:91;
  consider x,y,w,z be Element of REAL such that
A2: c = [*x,y,w,z*] by Lm1;
  1q * c = [* 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 thesis by A2;
end;
