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

theorem Th31:
  r <> 0 implies r * r" = 1
proof
  assume
A1: r <> 0;
  consider r0,r1,r2,r3 being Element of REAL such that
A2: r = [*r0,r1,r2,r3*] by Lm1;
A3: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*1,0,0,0*] by QUATERNI:91;
A4: Rea r = r0 by A2,QUATERNI:23;
A5: Im1 r = r1 by A2,QUATERNI:23;
A6: Im2 r = r2 by A2,QUATERNI:23;
A7: Im3 r = r3 by A2,QUATERNI:23;
  0 <= (Rea r)^2 + (Im1 r)^2 + (Im2 r)^2 + (Im3 r)^2 by QUATERNI:74; then
A8: |.r.|^2 = r0^2 + r1^2 + r2^2 + r3^2 by A4,A5,A6,A7,SQUARE_1:def 2;
A9: r"=[* (r0*1+r1*0+r2*0+r3*0)/(|.r.|^2), (r0*0-r1*1-r2*0+r3*0)/(|.r.|^2),
  (r0*0+r1*0-r2*1-r3*0)/(|.r.|^2),
  (r0*0-r1*0+r2*0-r3*jj)/(|.r.|^2) *] by A2,A3,Def1,Lm5
    .=[* (r0)/(|.r.|^2),(-r1)/(|.r.|^2),
  (-r2)/(|.r.|^2),(-r3)/(|.r.|^2) *];
  |.r.| <> 0 by A1,Th10; then
A10: |.r.|^2 > 0 by SQUARE_1:12;
  r*r"=[*r0*(r0/(|.r.|^2))-r1*((-r1)/(|.r.|^2))-
  r2*((-r2)/(|.r.|^2))-r3*((-r3)/(|.r.|^2)),
  r0*((-r1)/(|.r.|^2))+r1*(r0/(|.r.|^2))+
  r2*((-r3)/(|.r.|^2))-r3*((-r2)/(|.r.|^2)),
  r0*((-r2)/(|.r.|^2))+(r0/(|.r.|^2))*r2+
  ((-r1)/(|.r.|^2))*r3-((-r3)/(|.r.|^2))*r1,
  r0*((-r3)/(|.r.|^2))+r3*(r0/(|.r.|^2))+
  r1*((-r2)/(|.r.|^2))-r2*((-r1)/(|.r.|^2))
  *] by A2,A9,QUATERNI:def 10
    .=[*|.r.|^2/|.r.|^2,0,0,0 *] by A8
    .=[*1,0,0,0 *] by A10,XCMPLX_1:60
    .=[*jj,In(0,REAL)*] by QUATERNI:91
    .=1 by ARYTM_0:def 5;
  hence thesis;
end;
