
theorem Th5:
  for x,y being Real st x+y <> 0 holds (x+y)|^ 4 = (x|^ 3 +
  ((3*y)*x^2+(3*y^2)*x) +y|^ 3)*x + (x|^ 3 + ((3*y)*x^2+(3*y^2)*x) +y|^ 3)*y
proof
  let x,y be Real;
  assume
A1: x+y <> 0;
  per cases by A1,XXREAL_0:1;
  suppose
A2: x+y>0;
    (x+y)|^ 4 = (x+y) to_power (3+1) by POWER:41
      .= ((x+y) to_power 3)*((x+y) to_power 1) by A2,POWER:27
      .= ((x+y) to_power 3)*(x+y);
    then (x+y)|^ 4 = ((x+y)|^ 3)*(x+y)
      .= (x|^ 3 +((3*y)*x^2+(3*y^2)*x)+y|^ 3)*(x+y) by POLYEQ_1:14;
    hence thesis;
  end;
  suppose
    x+y<0;
    then
A3: -(x+y)>0 by XREAL_1:58;
    (-(x+y))|^ 4 = (x+y)|^ 4 by Lm1,POWER:1;
    then (x+y)|^ 4 = (-(x+y)) to_power (3+1) by POWER:41
      .= ((-(x+y)) to_power 3)*((-(x+y)) to_power 1) by A3,POWER:27
      .= ((-(x+y))|^ 3)*((-(x+y)) to_power 1);
    then
 (x+y)|^ 4 = ((-(x+y))|^ 3)*(-(x+y));
    then (x+y)|^ 4 = (-((x+y)|^ 3))*(-(x+y)) by Lm2,POWER:2
      .= ((x+y)|^ 3)*(x+y)
      .= (x|^ 3 +((3*y)*x^2+(3*y^2)*x)+y|^ 3)*(x+y) by POLYEQ_1:14;
    hence thesis;
  end;
end;
