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 Th13:
  z = x - a2/3 & q = (3*a1 - a2|^2)/9 & r = (9*a2*a1 - 2*a2|^3 -
  27*a0)/54 implies (z|^3+a2*z|^2+a1*z+a0 = 0 iff x|^3+3*q*x-2*r = 0)
proof
  assume
A1: z = x - a2/3;
  then
A2: 3*z = 3*x - a2;
A3: (3*x)|^2 = 3|^2*x|^2 by NEWTON:7
    .= (3*3)*x|^2 by Th1
    .= 9*x|^2;
A4: (3*x)|^3 = (3|^3)*(x|^3) by NEWTON:7
    .= (3*3*3)*x|^3 by Th2
    .= 27*x|^3;
A5: 27*z|^3 = 3*3*3*z|^3 .= 3|^3*z|^3 by Th2
    .= ((3*x) - a2)|^3 by A2,NEWTON:7
    .= 27*x|^3-3*(9*x|^2)*a2+3*a2|^2*(3*x)-a2|^3 by A4,A3,Th5
    .= 27*x|^3-27*a2*x|^2+9*a2|^2*x-a2|^3;
  assume
A6: q = (3*a1 - a2|^2)/9 & r = (9*a2*a1 - 2*a2|^3 - 27*a0)/54;
A7: 27*a1*z = 27*a1*x -9*a2*a1 by A1;
  (27*1)*a2*z|^2 = 3*a2*(3*3)*(1*1)*z|^2 .= 3*a2*3|^2*z|^2 by Th1
    .= 3*a2*(3|^2*z|^2)
    .= 3*a2*(3*z)|^2 by NEWTON:7
    .= 3*a2*((3*x)|^2 -2*(3*x)*a2 +a2|^2) by A2,Th4
    .= 3*a2*((3*x)*(3*x) -2*(3*x)*a2 +a2|^2) by Th1
    .= 27*a2*x*x -2*(3*x)*a2*(3*a2) +a2|^2*(3*a2)
    .= 27*a2*(x*x) -18*(a2*a2)*x +3*a2*(a2*a2) by Th1
    .= 27*a2*x|^2 -18*(a2*a2)*x +3*a2*a2*a2 by Th1
    .= 27*a2*x|^2 -18*a2|^2*x +3*(a2*a2*a2) by Th1
    .= 27*a2*x|^2 -18*a2|^2*x +3*a2|^3 by Th2;
  then
  27*(z|^3+a2*z|^2+a1*z+a0) = (27*x|^3)+(-9*a2|^2+27*a1)*x +(2*a2|^3-9*a2*
  a1+27*a0) by A5,A7
    .= 27*(x|^3 +3*q*x -2*r) by A6;
  hence thesis;
end;
