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 Th21:
  z = x - a3/4 & p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3|^
3)/64 & r = (256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024 implies (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)
proof
  assume
A1: z = x - a3/4;
  then
A2: 4*z = 4*x - a3;
A3: 16*z|^2 = (4*4)*z|^2 .= (4|^2*z|^2) by Th1
    .= (4*z)|^2 by NEWTON:7
    .= (4*x)|^2 -2*(4*x)*a3 +a3|^2 by A2,Th4
    .= (4*x)*(4*x) -2*(4*x)*a3 +a3|^2 by Th1
    .= 16*(x*x) -8*x*a3 +a3|^2
    .= 16*x|^2 -8*x*a3 +a3|^2 by Th1;
A4: (4*x)|^3 = (4|^3)*(x|^3) by NEWTON:7
    .= (4*4*4)*x|^3 by Th2
    .= 64*x|^3;
A5: (4*x)|^2 = 4|^2*x|^2 by NEWTON:7
    .= (4*4)*x|^2 by Th1
    .= 16*x|^2;
A6: 4*a3*(64*z|^3) = 4*a3*(4*4*4*z|^3) .= 4*a3*(4|^3*z|^3) by Th2
    .= 4*a3*((4*x) - a3)|^3 by A2,NEWTON:7
    .= 4*a3*(64*x|^3-3*(16*x|^2)*a3+3*a3|^2*(4*x)-a3|^3) by A4,A5,Th5
    .= 256*a3*x|^3-192*(a3*a3)*x|^2+48*(a3|^2*a3)*x-4*(a3|^3*a3)
    .= 256*a3*x|^3-192*(a3|^2)*x|^2+48*(a3|^2*a3)*x-4*(a3|^3*a3) by Th1
    .= 256*a3*x|^3-192*a3|^2*x|^2+48*(a3|^(2+1))*x-4*(a3|^3*a3) by NEWTON:6
    .= 256*a3*x|^3-192*a3|^2*x|^2+48*(a3|^3)*x-4*(a3|^(3+1)) by NEWTON:6
    .= 256*a3*x|^3-192*a3|^2*x|^2+48*a3|^3*x-4*a3|^4;
A7: (4*x)|^4 = (4|^4)*(x|^4) by NEWTON:7
    .= (4*4*4*4)*x|^4 by Th3
    .= 256*x|^4;
A8: 256*z|^4 = 4*4*4*4*z|^4 .= 4|^4*z|^4 by Th3
    .= ((4*x) - a3)|^4 by A2,NEWTON:7
    .= 256*x|^4-4*(64*x|^3)*a3+6*(16*x|^2)*a3|^2-4*a3|^3*(4*x)+a3|^4 by A7,A4
,A5,Th6
    .= 256*x|^4-256*x|^3*a3+96*x|^2*a3|^2-16*a3|^3*x+a3|^4;
  assume
A9: p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3|^3)/64 & r = (256*a0
  -64 *a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024;
  256*(z|^4+a3*z|^3+a2*z|^2+a1*z+a0) = 256*z|^4+4*a3*(64*z|^3)+16*a2*(16*
  z|^2)+64*a1*(4*z)+256*a0
    .= (256*x|^4-256*x|^3*a3+96*x|^2*a3|^2-16*a3|^3*x+a3|^4) +(256*a3*x|^3-
192*a3|^2*x|^2+48*a3|^3*x-4*a3|^4) +16*a2*(16*x|^2 -8*x*a3 +a3|^2)+64*a1*(4*x -
  a3)+256*a0 by A1,A8,A6,A3
    .= 256*(x|^4+4*p*x|^2+8*q*x+4*r) by A9;
  hence thesis;
end;
