reserve z1,z2,z3,z4,z for Quaternion;

theorem
  (-1q)^2=1
proof
  1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*jj,0,0,0*] by QUATERNI:91;
 then A1: -1q=[*-jj,-0,-0,-0*] by QUATERN2:4;
 then A2: Rea -1q =-1 & Im1 -1q= 0 by QUATERNI:23;
A3: Im2 -1q = 0 & Im3 -1q = 0 by A1,QUATERNI:23;
 (-1q)^2=[*(Rea -1q)^2-(Im1 -1q)^2-(Im2 -1q)^2-(Im3 -1q)^2, 2*(Rea (-1q)
  * Im1 (-1q)), 2*(Rea (-1q) * Im2 (-1q)), 2*(Rea (-1q) * Im3 (-1q))*] by Th78
    .=[*jj,In(0,REAL)*] by A2,A3,QUATERNI:91
    .=1 by ARYTM_0:def 5;
  hence thesis;
end;
