
theorem
  for x,y being Real st x+y <> 0 holds (x+y)|^ 4 = x|^ 4+((4*y)*(
  x|^ 3)+6*y^2*x^2+4*(y|^ 3)*x)+y|^ 4
proof
  let x,y be Real;
  set g = (x|^ 3 +((3*y)*x^2+(3*y^2)*x)+y|^ 3)*x;
  set h = (x|^ 3 +((3*y)*x^2+(3*y^2)*x)+y|^ 3)*y;
  set p = y|^ 3, q = x|^ 3, u = x|^ 4, v = y|^ 4;
A1: g = (x|^ 3)*x +((3*y)*x^2+(3*y^2)*x)*x+(y|^ 3)*x;
A2: y|^ 3 = y^2*y by Th4;
  assume x+y<>0;
  then
A3: (x+y)|^ 4 = g+h by Th5;
  h = (x|^ 3)*y +((3*y)*x^2+(3*y^2)*x)*y+((y|^ 3)*y)
    .= (x|^ 3)*y +((3*y)*x^2+(3*y^2)*x)*y+(y|^ 4) by Th4;
  then
  (x+y)|^ 4 = u +((3*y)*(x^2*x)-(-(3*y^2)*x)*x)+p*x +(q*y +((3*y)*x^2+(3*y
  ^2)*x)*y+v) by A3,A1,Th4
    .= u +((3*y)*q-(-((3*y^2)*x^2)))+p*x +(q*y +((3*y)*x^2+(3*y^2)*x)*y+v)
  by Th4
    .= u +(3*y)*q+(((3*y^2)*x^2)+p*x) +(q*y +(((3*y^2)*x^2)+(3*y^2)*x*y)+v);
  hence thesis by A2;
end;
