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

theorem
  (-1q)^3=-1
proof
A1: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*1,0,0,0*] by QUATERNI:91; then
A2: -1q=[*-jj,-0,-0,-0*] by QUATERN2:4; then
A3: Rea -1q =-1 & Im1 -1q= 0 by QUATERNI:23;
A4: Im2 -1q = 0 & Im3 -1q = 0 by A2,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 A3,A4,QUATERNI:91
    .=1 by ARYTM_0:def 5;
  then (-1q)^3=-1q by QUATERN2:15
    .=[*-jj,-0,-0,-0*] by A1,QUATERN2:4
    .=[*-jj,-In(0,REAL)*] by QUATERNI:91
    .=-1 by ARYTM_0:def 5;
  hence thesis;
end;
