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 Th14:
  a2 = -(s1+s2+s3) & a1 = s1*s2+s1*s3+s2*s3 & a0 = -s1*s2*s3
  implies (z|^3 + a2*z|^2 + a1*z + a0 = 0 iff z = s1 or z = s2 or z = s3)
proof
  assume a2 = -(s1+s2+s3) & a1 = s1*s2+s1*s3+s2*s3 & a0 = -s1*s2*s3;
  then
A1: (z-s1)*(z-s2)*(z-s3) = z*z*z + a2*z*z +a1*z +a0
    .= z|^3 + a2*(z*z) + a1*z + a0 by Th2
    .= z|^3 + a2*z|^2 + a1*z + a0 by Th1;
  hereby
    assume z|^3 + a2*z|^2 + a1*z + a0 = 0;
    then
A2: (z-s1)*(z-s2) = 0 or z-s3 = 0 by A1;
    assume ( not z = s1)& not z = s2;
    then z-s1<>0 & z-s2<>0;
    hence z = s3 by A2;
  end;
  assume z = s1 or z = s2 or z = s3;
  hence z|^3 + a2*z|^2 + a1*z + a0 = 0 by A1;
end;
