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

theorem Th33:
  c <> 0q implies c/c = 1q
proof
  assume
A1: c <> 0q;
  consider x1,x2,x3,x4 be Element of REAL such that
A2: c = [*x1,x2,x3,x4*] by Lm1;
  |.c.| > 0 by A1,Th10; then
A3: |.c.|^2 > 0 by SQUARE_1:12;
A4: Rea c = x1 by A2,QUATERNI:23;
A5: Im1 c = x2 by A2,QUATERNI:23;
A6: Im2 c = x3 by A2,QUATERNI:23;
A7: Im3 c = x4 by A2,QUATERNI:23;
A8: x1^2+x2^2+x3^2+x4^2 >= 0 by Lm2;
  c/c =[* (x1*x1+x2*x2+x3*x3+x4*x4)/(|.c.|^2),
  (x1*x2-x2*x1-x3*x4+x4*x3)/(|.c.|^2),
  (x1*x3+x2*x4-x3*x1-x4*x2)/(|.c.|^2),
  (x1*x4-x2*x3+x3*x2-x4*x1)/(|.c.|^2) *] by A2,Def1
    .=[* |.c.|^2/(|.c.|^2),0,0,0 *] by A4,A5,A6,A7,A8,SQUARE_1:def 2
    .=[* 1,0,0,0 *] by A3,XCMPLX_1:60
    .=[* jj,In(0,REAL) *] by QUATERNI:91
    .=1 by ARYTM_0:def 5;
  hence thesis;
end;
