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

theorem
  0q^3 = 0
proof
A1: 0q=[*In(0,REAL),In(0,REAL)*] by ARYTM_0:def 5
    .= [*0,0,0,0*] by QUATERNI:91; then
A2: Rea 0q =0 & Im1 0q= 0 by QUATERNI:23;
A3: Im2 0q = 0 & Im3 0q = 0 by A1,QUATERNI:23;
  0q^2=[*(Rea 0q)^2-(Im1 0q)^2-(Im2 0q)^2-(Im3 0q)^2, 2*(Rea 0q * Im1 0q),
  2*(Rea 0q * Im2 0q), 2*(Rea 0q * Im3 0q)*] by Th78
    .=[*In(0,REAL),In(0,REAL)*] by A2,A3,QUATERNI:91
    .=0 by ARYTM_0:def 5;
  hence thesis;
end;
