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

theorem Th1:
  b/a<0 & c/a>0 & delta(a,b,c) >=0 implies (-b+sqrt delta(a,b,c))/(
  2*a)>0 & (-b-sqrt delta(a,b,c))/(2*a)>0
proof
  assume that
A1: b/a < 0 and
A2: c/a>0 and
A3: delta(a,b,c) >=0;
A4: b^2-4*a*c >= 0 by A3,QUIN_1:def 1;
  now
    per cases by A1,XREAL_1:143;
    case
A5:   b<0 & a>0;
A6:   0 <= sqrt( b^2-4*a*c) by A4,SQUARE_1:17,26;
A7:   2*a>0 by A5,XREAL_1:129;
      -b>0 by A5,XREAL_1:58;
      then -b+sqrt(b^2-4*a*c)>0 +0 by A6;
      then
A8:   (-b+sqrt(b^2-4*a*c))/(2*a)>0 by A7,XREAL_1:139;
      c>0 & 4*a>0 by A2,A5,XREAL_1:129;
      then --4*a*c>0 by XREAL_1:129;
      then -4*a*c<0;
      then b^2+(-4*a*c)<b^2+0 by XREAL_1:8;
      then sqrt(b^2-4*a*c)<sqrt(b^2) by A4,SQUARE_1:27;
      then sqrt(b^2-4*a*c)<-b by A5,SQUARE_1:23;
      then -sqrt(b^2-4*a*c)>-(-b) by XREAL_1:24;
      then -sqrt(b^2-4*a*c)+(-b)>-(-b)+(-b) by XREAL_1:8;
      then (-b-sqrt(b^2-4*a*c))/(2*a)>0 by A7,XREAL_1:139;
      hence thesis by A8,QUIN_1:def 1;
    end;
    case
A9:   b>0 & a<0;
      then
A10:  a*2<0*2 by XREAL_1:68;
      c<0 by A2,A9;
      then a*c>0 by A9,XREAL_1:130;
      then 4*(a*c)>0 by XREAL_1:129;
      then --4*a*c>0;
      then -4*a*c<0;
      then b^2+(-4*a*c)<b^2+0 by XREAL_1:8;
      then sqrt(b^2-4*a*c)<sqrt(b^2) by A4,SQUARE_1:27;
      then sqrt(b^2-4*a*c)< b by A9,SQUARE_1:22;
      then -b+sqrt(b^2-4*a*c)<0+b+-b by XREAL_1:8;
      then
A11:  (-b+sqrt(b^2-4*a*c))/(2*a)>0 by A10,XREAL_1:140;
A12:  0 <= sqrt( b^2-4*a*c) by A4,SQUARE_1:17,26;
      --b>0 by A9;
      then -b+0<0+sqrt( b^2-4*a*c) by A12;
      then --(sqrt( b^2-4*a*c)+b)>0 by XREAL_1:62;
      then -b-sqrt( b^2-4*a*c)<0;
      then (-b-sqrt( b^2-4*a*c))/(2*a)>0 by A10,XREAL_1:140;
      hence thesis by A11,QUIN_1:def 1;
    end;
  end;
  hence thesis;
end;
