reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;
reserve a0,a1,a2,s1,s2 for Complex;
reserve a3,x,q,r,s,s3 for Complex;

theorem
  q = (3*a1 - a2|^2)/9 & q = 0 & r = (9*a2*a1 - 2*a2|^3 - 27*a0)/54 & s1
= 3-root(2*r) implies ( z|^3+a2*z|^2+a1*z+a0 = 0 iff z = s1-a2/3 or z = -s1/2-
  a2/3+s1*(2-root 3)*<i>/2 or z = -s1/2-a2/3-s1*(2-root 3)*<i>/2)
proof
  assume that
A1: q = (3*a1 - a2|^2)/9 and
A2: q = 0;
  set t = s1/2;
  set x1 = 2*t, x2 = -t+t*(2-root 3)*<i>, x3 = -t-t*(2-root 3)*<i>;
A3: x1*x2+x1*x3+x2*x3 = -3*t*t-t*t*((2-root 3)*(2-root 3))*(-1)
    .= -3*t*t-t*t*((2-root 3)|^2)*(-1) by Th1
    .= -3*t*t-t*t*3*(-1) by Th7
    .= 3*q by A2;
  set x = z + a2/3;
  assume
A4: r = (9*a2*a1 - 2*a2|^3 - 27*a0)/54;
  assume
A5: s1 = 3-root(2*r);
A6: -(x1+x2+x3) = 0;
A7: s1*s1*s1 = s1|^3 by Th2
    .= 2*r by A5,Th7;
  -x1*x2*x3 = (-2*t)*(t*t-t*t*((2-root 3)*(2-root 3))*(-1))
    .= (-2*t)*(t*t-t*t*((2-root 3)|^2)*(-1)) by Th1
    .= (-2*t)*(t*t-t*t*3*(-1)) by Th7
    .= -2*r by A7;
  then
A8: x|^3+0*x|^2+(3*q)*x+(-2*r) = 0 iff x = x1 or x = x2 or x = x3 by A6,A3,Th14
;
  z = x - a2/3;
  then z|^3+a2*z|^2+a1*z+a0 = 0 iff x|^3+3*q*x-2*r = 0 by A1,A4,Th13;
  hence thesis by A8;
end;
