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

theorem Th1:
  for x,y,z,w being Real holds [*x,y,z,w*] = x + y*<i> + z*<j> + w*<k>
proof
  let x,y,z,w be Real;
  reconsider x0=x+0, y0=y+0 as Element of REAL by XREAL_0:def 1;
  <i> = [*In(0,REAL),jj*] by ARYTM_0:def 5;
  then <i> = [*0,jj,0,0*] by QUATERNI:91;
  then
A1: y*<i> = [*y*0,y*jj,y*0,y*0*] by QUATERNI:def 21
    .= [*0,y,0,0*];
A2: z*<j> = [*z*0,z*0,z*jj,y*0*] by QUATERNI:def 21
    .= [*0,0,z,0*];
A3: w*<k> = [*w*0,w*0,w*0,w*jj*] by QUATERNI:def 21
    .= [*0,0,0,w*];
  x + y* <i> = [*x0,y0,0,0*] by A1,QUATERNI:def 19
    .= [*x,y,0,0*];
  then x + y*<i> + z*<j> = [*x0,y0,0+z,0+0*] by A2,QUATERNI:def 7
    .= [*x,y,z,0*];
  then x + y*<i> + z*<j> + w*<k> = [*x0,y0,0+z,0+w*] by A3,QUATERNI:def 7;
  hence thesis;
end;
