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

theorem
  c1.|.(c2+c3) = c1.|.c2 + c1.|.c3
proof
  thus c1.|.(c2+c3) = c1 * (c2*' + c3*') by QUATERNI:54
    .= c1.|.c2 + c1.|.c3 by Th17;
end;
