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 Th49:
  c .|. c = |.c.|^2
proof
  reconsider z=0, z1=(Rea c)^2+(Im1 c)^2+(Im2 c)^2+(Im3 c)^2
   as Element of REAL by XREAL_0:def 1;
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
    .=[*z1,z*] by QUATERNI:91
    .= (Rea c)^2+(Im1 c)^2+(Im2 c)^2+(Im3 c)^2 by ARYTM_0:def 5
    .= |.c.|^2 by A1,SQUARE_1:def 2;
  hence thesis;
end;
