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 Th15:
  a <> 0 & Polynom(a,b,c,d,x) = 0 implies for a1,a2,a3,h,y st y =
(x+b/(3*a)) & h = -b/(3*a) & a1 = b/a & a2 = c/a & a3 = d/a holds y |^ 3 + ((3*
  h+a1)*y^2+(3*h^2+2*(a1*h)+a2)*y) + ((h |^ 3+a1*h^2)+(a2*h+a3)) = 0
proof
  assume
A1: a <> 0;
  assume
A2: Polynom(a,b,c,d,x) = 0;
  let a1,a2,a3,h,y;
  assume that
A3: y = x+ b/(3*a) & h = -b/(3*a) and
A4: a1 = b/a & a2 = c/a & a3 = d/a;
  0 = a"*(a*(x |^ 3)+ b*x^2 +(c*x +d)) by A2
    .= (a"*a)*(x |^ 3)+ a"*(b*x^2) +a"*(c*x +d)
    .= 1*(x |^ 3)+ a"*(b*x^2) +a"*(c*x +d) by A1,XCMPLX_0:def 7
    .= (x |^ 3)+ (a"*b)*x^2 +(a"*c)*x +a"*d
    .= (x |^ 3)+ (b/a)*x^2 +(a"*c)*x +a"*d by XCMPLX_0:def 9
    .= (x |^ 3)+ (b/a)*x^2 +(c/a)*x +a"*d by XCMPLX_0:def 9
    .= (x |^ 3)+ a1*x^2 + a2*x + a3 by A4,XCMPLX_0:def 9;
  then
  0 = y |^ 3 +((3*h)*y^2+(3*h^2)*y)+h |^ 3 + (a1*y^2+2*(a1*h)*y+a1*h^2) +
  a2*(y+h) + a3 by A3,Th14
    .= y |^ 3 +((3*h)*y^2+(3*h^2)*y)+(2*(a1*h)*y+a1*y^2) +(a2*y + ((h |^ 3 +
  a1*h^2)+(a2*h + a3)));
  hence thesis;
end;
