reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem
  0 <= a & c < 0 & delta(a,b,c) = 0 implies a = 0
  proof
    assume that
A1: 0 <= a and
A2: c < 0 and
A3: delta(a,b,c) = 0;
A4: b^2 - 4 * a * c = 0 by A3,QUIN_1:def 1;
    0 <= b^2
    proof
      per cases;
      suppose b = 0;
        hence thesis;
      end;
      suppose 0 <> b;
        hence thesis by SQUARE_1:12;
      end;
    end;
    hence thesis by A1,A2,A4;
  end;
