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 & 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 + 0*y^2 + ((
  3*a*c-b^2)/(3*a^2))*y + (2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2)) = 0
proof
  assume
A1: a <> 0;
  then
A2: 3*a <> 0;
  assume
A3: Polynom(a,b,c,d,x) = 0;
  let a1,a2,a3,h,y;
  assume that
A4: y = (x + b/(3*a)) and
A5: h = -b/(3*a) and
A6: a1 = b/a and
A7: a2 = c/a and
A8: a3 = d/a;
  set p0 = 3*h+a1;
A9: p0 = -(3*(b/(3*a)))+a1 by A5
    .= -(3*((3*a)"*b))+a1 by XCMPLX_0:def 9
    .= -(3*((3"*a")*b))+a1 by XCMPLX_1:204
    .= -(((3*3")*a")*b)+a1
    .= -(b/a)+a1 by XCMPLX_0:def 9;
  set q2 = (h |^ 3 +a1*h^2)+(a2*h + a3);
A10: q2 = 2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2)
  proof
    set t = 3*a;
    set a6 = b/t;
A11: b/a = (3*b)/t by XCMPLX_1:91;
A12: a6 |^ 3 = a6 |^ (2+1) .= (a6 |^ (1+1))*a6 by NEWTON:6
      .= (a6 |^ 1)*a6*a6 by NEWTON:6
      .= (a6 |^ 1)*a6^2
      .= (a6 to_power 1)*a6^2 by POWER:41;
    q2 = ((-b/t) |^ 3 +b/a*(-b/t)^2) +(-(c/a*(b/t)) + d/a) by A5,A6,A7,A8
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(-(b*c)/(t*a) + d/a) by XCMPLX_1:76
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(-(b*c)/(3*a^2) + 1*(d/a))
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(-(b*c)/(3*a^2) + (t/t)*(d/a)) by A2,
XCMPLX_1:60
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) + ((t/t)*(d/a))- (b*c)/(3*a^2)
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) + ((t*d)/(t*a))- (b*c)/(3*a^2) by
XCMPLX_1:76
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(t*d)*(3*a^2)"-(b*c)/(3*a^2) by
XCMPLX_0:def 9
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(t*d)*(3*a^2)"-(b*c)*(3*a^2)" by
XCMPLX_0:def 9
      .= ((-b/t) |^ 3 +b/a*(-b/t)^2) +(t*d-b*c)*(3*a^2)"
      .= (-b/t) |^ (2+1) +b/a*(b/t)^2 +(t*d-b*c)/(3*a^2) by XCMPLX_0:def 9
      .= ((-b/t) |^ (1+1))*(-b/t) +b/a*(b/t)^2 +( t*d-b*c)/(3*a^2) by NEWTON:6
      .= ((-b/t) |^ 1)*(-b/t)*(-b/t) +b/a*(b/t)^2+( t*d-b*c)/(3*a^2) by
NEWTON:6
      .= ((-b/t) |^ 1)*(-b/t)^2 +b/a*(b/t)^2 +( t*d-b*c)/(3*a^2)
      .= ((-b/t) to_power 1)*(-b/t)^2 +b/a*(b/t)^2 +( t*d-b*c)/(3*a^2) by
POWER:41
      .= (-b/t)*(b/t)^2 +b/a*(b/t)^2 +(t*d-b*c)/(3*a^2) by POWER:25
      .= (-b/t)*(b^2/t^2) +b/a*(b/t)^2 +(t*d-b*c)/(3*a^2) by XCMPLX_1:76
      .= ((-b/t)*(b^2/t^2) +b/a*(b^2/t^2)) +(t*d-b*c)/(3*a^2) by XCMPLX_1:76
      .= (( b/a -b/t)*(b^2/t^2)) +(t*d-b*c)/(3*a^2)
      .= (((3*b)*t" -1*b/t)*(b^2/t^2)) +( t*d-b*c)/(3*a^2) by A11,
XCMPLX_0:def 9
      .= (((3*b)*t" -1*b*t")*(b^2/t^2)) +( t*d-b*c)/(3*a^2) by XCMPLX_0:def 9
      .= 2*((b*t")*(b^2/t^2)) +(t*d-b*c)/(3*a^2)
      .= 2*((b/t)*(b^2/t^2)) +(t*d-b*c)/(3*a^2) by XCMPLX_0:def 9
      .= 2*((b/t)*(b/t)^2) +(t*d-b*c)/(3*a^2) by XCMPLX_1:76;
    hence thesis by A12,POWER:25;
  end;
  set p2 = (3*h^2+2*(a1*h)+a2);
A13: p2 = (3*a*c-b^2)/(3*a^2)
  proof
    set t = ((3*a)"*b);
A14: (3*a)/(3*a) = 1 by A2,XCMPLX_1:60;
    p2 = (3*(b/(3*a))^2+2*(a1*-b/(3*a))+a2) by A5
      .= (3*((3*a)"*b)^2+2*(a1*-b/(3*a))+a2) by XCMPLX_0:def 9
      .= ((3*(3*a)")*b)*t+2*(a1*-b/(3*a))+a2
      .= (3*(3"*a")*b)*t+2*(a1*-b/(3*a))+a2 by XCMPLX_1:204
      .= (b/a)*((3*a)"*b)+2*(a1*-b/(3*a))+a2 by XCMPLX_0:def 9
      .= (b/a)*(b/(3*a))+2*(a1*-b/(3*a))+a2 by XCMPLX_0:def 9
      .= (b*b)/(a*(3*a))+2*(a1*-b/(3*a))+a2 by XCMPLX_1:76
      .= b^2/(3*a^2)-2*(b/a*(b/(3*a))) +c/a by A6,A7
      .= b^2/(3*a^2)-2*((b*b)/(a*(3*a))) +c/a by XCMPLX_1:76
      .= -b^2/(3*a^2) +((3*a)/(3*a))*(c/a) by A14
      .= -b^2/(3*a^2) +(3*a*c)/(3*a*a) by XCMPLX_1:76
      .= (3*a*c)/(3*a^2)-b^2/(3*a^2)
      .= (3*a*c)*(3*a^2)"-b^2/(3*a^2) by XCMPLX_0:def 9
      .= (3*a*c)*(3*a^2)"-b^2*(3*a^2)" by XCMPLX_0:def 9
      .= ((3*a*c)-b^2)*(3*a^2)";
    hence thesis by XCMPLX_0:def 9;
  end;
  y |^ 3 +(p0*y^2+p2*y) + q2 = 0 by A1,A3,A4,A5,A6,A7,A8,Th15;
  hence thesis by A6,A9,A13,A10;
end;
