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

theorem
  1q^2 = 1
proof
A1: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*jj,0,0,0*] by QUATERNI:91; 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;
