reserve a, a9, a1, a2, a3, b, b9, c, c9, d, d9, h, p, q, x, x1, x2, x3, u, v,
  y, z for Real;

theorem
  a <> 0 & c/a < 0 & Polynom(a,0,c,0,x) = 0 implies x = 0 or x = sqrt -c
  /a or x = -sqrt(-c/a)
proof
  assume that
A1: a <> 0 and
A2: c/a < 0;
  x |^ 3 = x |^ (2+1);
  then x |^ 3 = (x |^ (1+1))*x by NEWTON:6;
  then
A3: x |^ 3 = ((x |^ 1)*x)*x by NEWTON:6;
A4: x |^ 3 = x^2*x by A3;
  assume Polynom(a,0,c,0,x) = 0;
  then (a*x^2+c)*x = 0 by A4;
  then
A5: x = 0 or (a*x^2+c) = 0 by XCMPLX_1:6;
  now
    per cases by XXREAL_0:1;
    case
A6:   x > 0;
      x |^ 2 = (x |^ (1+1));
      then x |^ 2 = ((x |^ 1)*x) by NEWTON:6;
      then x |^ 2 = ((x to_power 1)*x) by POWER:41;
      then
A7:   x |^ 2 = x*x by POWER:25;
A8:   (-c/a) > 0 by A2,XREAL_1:58;
      x^2 = (-c)/a by A1,A5,A6,XCMPLX_1:89;
      then x^2 = ((-c)*a") by XCMPLX_0:def 9;
      then x^2 = -(c*a");
      then x^2 = -c/a by XCMPLX_0:def 9;
      then x = 2 -Root (-c/a) by A6,A8,A7,PREPOWER:def 2;
      hence thesis by A8,PREPOWER:32;
    end;
    case
A9:   x < 0;
      (-x) |^ 2 = ((-x) |^ (1+1));
      then (-x) |^ 2 = (((-x) |^ 1)*(-x)) by NEWTON:6;
      then (-x) |^ 2 = (((-x) to_power 1)*(-x)) by POWER:41;
      then
A10:  (-x) |^ 2 = (-x)*(-x) by POWER:25;
      x^2 = (-c)/a by A1,A5,A9,XCMPLX_1:89;
      then x^2 = ((-c)*a") by XCMPLX_0:def 9;
      then x^2 = -(c*a");
      then
A11:  (-x)^2=-c/a by XCMPLX_0:def 9;
A12:  (-c/a) > 0 by A2,XREAL_1:58;
      -x>0 by A9,XREAL_1:58;
      then -x = (2 -Root (-c/a)) by A11,A12,A10,PREPOWER:def 2;
      then -x = sqrt(-c/a) by A12,PREPOWER:32;
      hence thesis;
    end;
    case
      x=0;
      hence thesis;
    end;
  end;
  hence thesis;
end;
