reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;
reserve a0,a1,a2,s1,s2 for Complex;

theorem Th11:
  a1 = -(s1+s2) & a0 = s1*s2 implies (z|^2 + a1*z + a0 = 0 iff z =
  s1 or z = s2)
proof
  assume a1 = -(s1+s2) & a0 = s1*s2;
  then
A1: (z-s1)*(z-s2) = z*z +a1*z +a0 .= z|^2 + a1*z + a0 by Th1;
  hereby
    assume z|^2 + a1*z + a0 = 0;
    then
A2: z-s1 = 0 or z-s2 = 0 by A1;
    assume not z = s1;
    hence z = s2 by A2;
  end;
  assume z = s1 or z = s2;
  hence z|^2 + a1*z + a0 = 0 by A1;
end;
