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

theorem
  z - 0q = z
proof
  consider x,y,w,v be Real such that
A1: z = [*x,y,w,v*] by QUATERNI:7;
  0q = [*In(0,REAL),In(0,REAL)*] by ARYTM_0:def 5
    .= [*0,0,0,0*] by QUATERNI:91;
  then -0q=[*-0,-0,-0,-0*] by QUATERN2:4;
  then z- 0q =[* x+(-0),y+(-0),w+(-0),v+(-0) *] by A1,QUATERNI:def 7
    .= [*x,y,w,v*];
  hence thesis by A1;
end;
