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 Th22:
  a3 = -(s1+s2+s3+s4) & a2 = s1*s2+s1*s3+s1*s4+s2*s3+s2*s4+s3*s4 &
a1 = -(s1*s2*s3+s1*s2*s4+s1*s3*s4+s2*s3*s4) & a0=s1*s2*s3*s4 implies (z|^4 + a3
  *z|^3 + a2*z|^2 + a1*z + a0 = 0 iff z = s1 or z = s2 or z = s3 or z = s4)
proof
  assume a3 = -(s1+s2+s3+s4) & a2 = s1*s2+s1*s3+s1*s4+s2*s3+s2*s4+s3*s4 & a1
  = -(s1*s2*s3+s1*s2*s4+s1*s3*s4+s2*s3*s4) & a0=s1*s2*s3*s4;
  then
A1: (z-s1)*(z-s2)*(z-s3)*(z-s4) = z*z*z*z + a3*z*z*z +a2*z*z +a1*z +a0
    .= z|^4 + a3*(z*z*z) + a2*z*z + a1*z + a0 by Th3
    .= z|^4 + a3*z|^3 + a2*(z*z) + a1*z + a0 by Th2
    .= z|^4 + a3*z|^3 + a2*z|^2 + a1*z + a0 by Th1;
  hereby
    assume z|^4 + a3*z|^3 + a2*z|^2 + a1*z + a0 = 0;
    then
A2: (z-s1)*(z-s2)*(z-s3) = 0 or z-s4 = 0 by A1;
    assume that
A3: ( not z = s1)& not z = s2 and
A4: not z = s3;
A5: z-s3<>0 by A4;
    z-s1<>0 & z-s2<>0 by A3;
    hence z = s4 by A2,A5;
  end;
  assume z = s1 or z = s2 or z = s3 or z = s4;
  hence z|^4 + a3*z|^3 + a2*z|^2 + a1*z + a0 = 0 by A1;
end;
