
theorem
  for a,k,y being Real st a <> 0 holds (for x being Real
  holds x|^ 4+a|^ 4 = k*a*x*(x^2+a^2)) implies y|^ 4 -k*(y|^ 3)-k*y+1 = 0
proof
  let a,k,y be Real;
  assume that
A1: a <> 0 and
A2: for x being Real holds x|^ 4+a|^ 4 = k*a*x*(x^2+a^2);
  (a*y)|^ 4+a|^ 4 = k*a*(a*y)*((a*y)^2+a^2) by A2
    .= k*(a^2*y)*(a^2*y^2+a^2*1);
  then (a*y)|^ 4+a|^ 4 = k*(((a^2*a^2)*y)*(y^2+1))
    .= (k*(((a|^ 4)*y)*(y^2+1))) by Th4
    .= (((a|^ 4)*(k*y))*(y^2+1));
  then (a|^ 4)*(y|^ 4)+(a|^ 4)*1 = (a|^ 4)*((k*y)*(y^2+1)) by NEWTON:7;
  then (a|^ 4)"*((a|^ 4)*((y|^ 4)+1-(k*y)*(y^2+1))) = 0;
  then (((a|^ 4)"*(a|^ 4))*((y|^ 4)+1-(k*y)*(y^2+1))) = 0;
  then
A3: (((1/(a|^ 4))*(a|^ 4))*((y|^ 4)+1-(k*y)*(y^2+1))) = 0 by XCMPLX_1:215;
  a|^ 4 <> 0 by A1,PREPOWER:5;
  then 1*((y|^4)+1-(k*y)*(y^2+1)) = 0 by A3,XCMPLX_1:106;
  then (y|^4)-k*(y^2*y)-k*y+1 = 0;
  hence thesis by Th4;
end;
