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
  c.|.c = 0 implies c = 0
proof
  assume c.|.c = 0;
  then
A1: |.c.|^2 = 0 by Th49;
  (Rea c)^2+(Im1 c)^2+(Im2 c)^2+(Im3 c)^2 >= 0 by Lm2;
  then
A2: |.c.|^2 = (Rea c)^2+(Im1 c)^2+(Im2 c)^2+(Im3 c)^2 by SQUARE_1:def 2;
  then
A3: Rea c = 0 by A1,Lm3;
A4: Im1 c = 0 by A1,A2,Lm3;
A5: Im2 c = 0 by A1,A2,Lm3;
  Im3 c = 0 by A1,A2,Lm3;
  then c = [*0,0,0,0*] by A3,A4,A5,QUATERNI:24
    .= [*In(0,REAL),In(0,REAL)*] by QUATERNI:91
    .= 0 by ARYTM_0:def 5;
  hence thesis;
end;
