reserve x,y,a,b,c,p,q for Real;
reserve m,n for Element of NAT;

theorem
  c/a<0 implies (-b+sqrt delta(a,b,c))/(2*a)>0 & (-b-sqrt delta(a,b,c))/
  (2*a)<0 or (-b+sqrt delta(a,b,c))/(2*a)<0 & (-b-sqrt delta(a,b,c))/(2*a)>0
proof
  assume
A1: c/a<0;
  now
    per cases by A1,XREAL_1:143;
    case
A2:   c>0 & a<0;
      then 4*a<4*0 by XREAL_1:68;
      then 4*a*c<0*c by A2,XREAL_1:68;
      then
A3:   -4*a*c>0 by XREAL_1:58;
      then b^2+(-4*a*c)>b^2+0 by XREAL_1:8;
      then
A4:   sqrt(b^2-4*a*c)>sqrt(b^2) by SQUARE_1:27,XREAL_1:63;
A5:   2*a<2*0 by A2,XREAL_1:68;
      -4*a*c+b^2>0+0 by A3,XREAL_1:8,63;
      then
A6:   --sqrt(b^2-4*a*c)>0 by SQUARE_1:17,27;
      then
A7:   -sqrt(b^2-4*a*c)<0;
      now
        per cases;
        suppose
A8:       b>=0;
          then -b<=-0;
          then -sqrt(b^2-4*a*c)+-b<0+0 by A7;
          then -b-sqrt(b^2-4*a*c)<0;
          then
A9:       -b-sqrt delta(a,b,c)<0 by QUIN_1:def 1;
          sqrt(b^2-4*a*c)>b by A4,A8,SQUARE_1:22;
          then -b+sqrt(b^2-4*a*c)>0+b+-b by XREAL_1:8;
          then (-b+sqrt(b^2-4*a*c))/(2*a)<0 by A5,XREAL_1:142;
          hence thesis by A5,A9,QUIN_1:def 1,XREAL_1:140;
        end;
        suppose
A10:      b<0;
          then sqrt(b^2-4*a*c)>-b by A4,SQUARE_1:23;
          then --(b+sqrt(b^2-4*a*c))>0 by XREAL_1:62;
          then -b-sqrt(b^2-4*a*c)<0;
          then
A11:      (-b-sqrt(b^2-4*a*c))/(2*a)>0 by A5,XREAL_1:140;
          -b>0 by A10,XREAL_1:58;
          then sqrt(b^2-4*a*c)+(-b)>0+0 by A6;
          then sqrt delta(a,b,c)+(-b)>0+0 by QUIN_1:def 1;
          hence thesis by A5,A11,QUIN_1:def 1,XREAL_1:142;
        end;
      end;
      hence thesis;
    end;
    case
A12:  c<0 & a>0;
      then 4*a>0 by XREAL_1:129;
      then 4*a*c<4*a*0 by A12,XREAL_1:68;
      then
A13:  -4*a*c>0 by XREAL_1:58;
      then b^2+(-4*a*c)>b^2+0 by XREAL_1:8;
      then
A14:  sqrt(b^2-4*a*c)>sqrt(b^2) by SQUARE_1:27,XREAL_1:63;
A15:  2*a>0 by A12,XREAL_1:129;
      -4*a*c+b^2>0+0 by A13,XREAL_1:8,63;
      then
A16:  --sqrt(b^2-4*a*c)>0 by SQUARE_1:17,27;
      then
A17:  -sqrt(b^2-4*a*c)<0;
      now
        per cases;
        suppose
A18:      b>=0;
          then -b<=-0;
          then -sqrt(b^2-4*a*c)+-b<0+0 by A17;
          then -b-sqrt(b^2-4*a*c)<0;
          then
A19:      -b-sqrt delta(a,b,c)<0 by QUIN_1:def 1;
          sqrt(b^2-4*a*c)>b by A14,A18,SQUARE_1:22;
          then -b+sqrt(b^2-4*a*c)>0+b+-b by XREAL_1:8;
          then (-b+sqrt(b^2-4*a*c))/(2*a)>0 by A15,XREAL_1:139;
          hence thesis by A12,A19,QUIN_1:def 1,XREAL_1:129,141;
        end;
        suppose
A20:      b<0;
          then sqrt(b^2-4*a*c)>-b by A14,SQUARE_1:23;
          then --(b+sqrt(b^2-4*a*c))>0 by XREAL_1:62;
          then -b-sqrt(b^2-4*a*c)<0;
          then
A21:      (-b-sqrt(b^2-4*a*c))/(2*a)<0 by A12,XREAL_1:129,141;
          -b>0 by A20,XREAL_1:58;
          then sqrt(b^2-4*a*c)+(-b)>0+0 by A16;
          then sqrt delta(a,b,c)+(-b)>0 by QUIN_1:def 1;
          hence thesis by A15,A21,QUIN_1:def 1,XREAL_1:139;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
