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).|.(c1 + c2) = c1.|.c1 + c1.|.c2 + c2.|.c1 + c2.|.c2
proof
  (c1 + c2).|.(c1 + c2)=(c1 + c2) * (c1*' + c2*') by QUATERNI:54
    .=(c1 + c2) * c1*' + (c1 + c2) * c2*' by Th17
    .=c1 * c1*' + c2 * c1*' + (c1 + c2) * c2*' by Th18
    .=c1.|.c1 + c2.|.c1 + (c1.|.c2 + c2.|.c2) by Th18
    .=c1.|.c1 + c2.|.c1 + c1.|.c2 + c2.|.c2 by Th2;
  hence thesis by Th2;
end;
