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 Th5:
  a <> 0 & delta(a,b,c) >= 0 implies for x holds Polynom(a,b,c,x) =
0 implies x = (-b+sqrt delta(a,b,c))/(2*a) or x = (-b-sqrt delta(a,b,c))/(2*a)
proof
  assume that
A1: a <> 0 and
A2: delta(a,b,c)>=0;
  now
    set e = a*c;
    let y;
    set t = a^2*y^2+(a*b)*y;
    assume Polynom(a,b,c,y) = 0;
    then a*(a*y^2+b*y+c) = a*0;
    then a*(a*y^2)+a*(b*y)+a*c = 0;
    then
A3: t +b^2/4-b^2*4" = -(4*e)*4";
A4: delta(a,b,c) = b^2-4*a*c by QUIN_1:def 1;
A5: now
      assume ((a*y+b/2) -sqrt((b^2-4*(a*c))/4)) =0;
      then (a*y+b/2) = sqrt(b^2-4*(a*c))/2 by A2,A4,SQUARE_1:20,30;
      then a*y = -(b*2" - sqrt(b^2-4*(a*c))*2" )
        .= ((-b)*2" +(sqrt delta(a,b,c)*2")) by A4;
      then y = ((-b)*2" +(sqrt delta(a,b,c)*2")) /a by A1,XCMPLX_1:89
        .= ((-b)*2" +(sqrt delta(a,b,c)*2"))*a" by XCMPLX_0:def 9
        .= (-b +sqrt delta(a,b,c))*(2"*a")
        .= (-b +sqrt delta(a,b,c))*(2*a)" by XCMPLX_1:204;
      hence y = (-b +sqrt delta(a,b,c))/(2*a) by XCMPLX_0:def 9;
    end;
A6: now
      assume (a*y+b/2) +sqrt((b^2-4*(a*c))/4) = 0;
      then (a*y+b/2) = - sqrt((b^2-4*(a*c))/4);
      then a*y+b/2 = -sqrt(b^2-4*(a*c))/2 by A2,A4,SQUARE_1:20,30;
      then a*y = -(b*2" + sqrt(b^2-4*(a*c))*2" )
        .= ((-b)*2" -(sqrt delta(a,b,c)*2")) by A4;
      then y = ((-b)*2" -(sqrt delta(a,b,c)*2")) /a by A1,XCMPLX_1:89
        .= ((-b)*2" -(sqrt delta(a,b,c)*2"))*a" by XCMPLX_0:def 9
        .= (-b -sqrt delta(a,b,c))*(2"*a")
        .= (-b -sqrt delta(a,b,c))*(2*a)" by XCMPLX_1:204;
      hence y = (-b -sqrt delta(a,b,c))/(2*a) by XCMPLX_0:def 9;
    end;
    (b^2-4*(a*c))/4 >= 0/4 by A2,A4,XREAL_1:72;
    then (a*y+b/2)^2 = (sqrt((b^2-4*(a*c))/4))^2 by A3,SQUARE_1:def 2;
    then
    ((a*y+b/2) - sqrt((b^2-4*(a*c))/4))* ((a*y+b/2) + sqrt((b^2-4*(a*c))/
    4)) = 0;
    hence y = (-b+sqrt delta(a,b,c))/(2*a) or y = (-b-sqrt delta(a,b,c))/(2*a)
    by A5,A6,XCMPLX_1:6;
  end;
  hence thesis;
end;
