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;
reserve a4,p,s4 for Complex;

theorem
  p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3|^3)/64 & q = 0 & r = (
256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024 & s1 = 2-root(p|^2-r) implies ( z
|^4+a3*z|^3+a2*z|^2+a1*z+a0 = 0 iff z = 2-root(-2*(p-s1))-a3/4 or z = -2-root(-
  2*(p-s1))-a3/4 or z = 2-root(-2*(p+s1))-a3/4 or z = -2-root(-2*(p+s1))-a3/4)
proof
  assume
A1: p = (8*a2-3*a3|^2)/32;
  set x = z + a3/4;
  assume that
A2: q = (8*a1 -4*a2*a3 + a3|^3)/64 and
A3: q = 0;
A4: z = x - a3/4;
  assume r = (256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024;
  then
A5: z|^4+a3*z|^3+a2*z|^2+a1*z+a0 = 0 iff x|^4+4*p*x|^2+8*q*x+4*r = 0 by A1,A2
,A4,Th21;
  assume
A6: s1 = 2-root(p|^2-r);
  set y = x|^2;
A7: y|^2 = x|^(2*2) by NEWTON:9
    .= x|^4;
A8: x|^2 = -2*p+2-root(delta(4*r,4*p,1))/2 iff x = 2-root(-2*p+2-root(delta
  (4*r,4*p,1))/2) or x = -2-root(-2*p+2-root(delta(4*r,4*p,1))/2) by Th10;
A9: x|^2 = -2*p-2-root(delta(4*r,4*p,1))/2 iff x = 2-root(-2*p-2-root(delta
  (4*r,4*p,1))/2) or x = -2-root(-2*p-2-root(delta(4*r,4*p,1))/2) by Th10;
A10: 4|^2 = 4*4 by Th1
    .= 16;
  1*y|^2+4*p*y+4*r = 0 iff y = -(4*p)/(2*1)+2-root(delta(4*r,4*p,1))/(2*1
  ) or y = -(4*p)/(2*1)-2-root(delta(4*r,4*p,1))/(2*1) by Th12;
  then
A11: 1*y|^2+4*p*y+4*r = 0 iff z = 2-root(-2*p+2-root(delta(4*r,4*p,1))/2)-a3
  /4 or z = -2-root(-2*p+2-root(delta(4*r,4*p,1))/2)-a3/4 or z = 2-root(-2*p-2
-root(delta(4*r,4*p,1))/2)-a3/4 or z = -2-root(-2*p-2-root(delta(4*r,4*p,1))/2)
  -a3/4 by A8,A9;
  2-root 16 = 2-real-root 16 by Th8
    .= 2-Root 16 by POWER:def 1
    .= 4 by A10,PREPOWER:def 2;
  then 2-root(delta(4*r,4*p,1))/4 = 2-root(16*(p*p-r)/16) by Th9
    .= s1 by A6,Th1;
  hence thesis by A3,A5,A7,A11;
end;
