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
  Rea (c .|. c) = |.c.|^2 & Im1 (c .|. c) = 0 &
  Im2 (c .|. c) = 0 & Im2 (c .|. c) = 0
proof
A1: (Rea c)^2+(Im1 c)^2+(Im2 c)^2+(Im3 c)^2 >= 0 by Lm2;
  c .|. c = [*(Rea c)*(Rea c)+(Im1 c)*(Im1 c)+(Im2 c)*(Im2 c)+(Im3 c)*(Im3 c),
  (Rea c)*(-(Im1 c))+(Im1 c)*(Rea c)-(Im2 c)*(Im3 c)+(Im3 c)*(Im2 c),
  (Rea c)*(-(Im2 c))+(Rea c)*(Im2 c)-(Im1 c)*(Im3 c)+(Im3 c)*(Im1 c),
  (Rea c)*(-(Im3 c))+(Im3 c)*(Rea c)-(Im1 c)*(Im2 c)+(Im2 c)*(Im1 c) *]
  by Th48
    .=[*|.c.|^2,0,0,0*] by A1,SQUARE_1:def 2;
  hence thesis by QUATERNI:23;
end;
