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(1_root_of_cubic(-q
|^2,p|^2-r,2*p)) & s2 = 2-root(2_root_of_cubic(-q|^2,p|^2-r,2*p)) & s3 = -q/(s1
*s2) implies ( z|^4+a3*z|^3+a2*z|^2+a1*z+a0 = 0 iff z = s1+s2+s3-a3/4 or z = s1
  -s2-s3-a3/4 or z = -s1+s2-s3-a3/4 or z = -s1-s2+s3-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;
  assume
A4: r = (256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024;
A5: z = x - a3/4;
  assume s1 = 2-root(1_root_of_cubic(-q|^2,p|^2-r,2*p)) & s2 = 2-root(
  2_root_of_cubic (-q|^2,p|^2-r,2*p)) & s3 = -q/(s1*s2);
  then
  x|^4+4*p*x|^2+8*q*x+4*r = 0 iff x = s1+s2+s3 or x = s1-s2-s3 or x = -s1+
  s2-s3 or x = -s1-s2+s3 by A3,Th23;
  hence thesis by A1,A2,A4,A5,Th21;
end;
