
theorem Th4:
  for x being Real holds x|^ 3 = x^2*x & (x|^ 3)*x = x|^ 4 &
  x^2*x^2 = x|^ 4
proof
  let x be Real;
  per cases by XXREAL_0:1;
  suppose
    x = 0;
    hence thesis by NEWTON:11;
  end;
  suppose
A1: x > 0;
    (x|^ 3)*x = (x|^ 3)*(x to_power 1)
      .= (x to_power 3)*(x to_power 1);
    then
A2: (x|^ 3)*x = x to_power (3 + 1) by A1,POWER:27;
    x^2 = x to_power 2 by POWER:46;
    then x^2*x = (x to_power 2)*(x to_power 1)
      .= x to_power (2 + 1) by A1,POWER:27
      .= x|^ 3 by POWER:41;
    hence thesis by A2,POWER:41;
  end;
  suppose
    x<0;
    then
A3: -x>0 by XREAL_1:58;
    (-x)|^ 3 + (x|^ 3) = -((x|^ 3)+-(x|^ 3)) by Lm2,POWER:2
      .= (x|^ 3)-(x|^ 3);
    then
A4: (x|^ 3)+ ((-x)|^ 3 - (-x)|^ 3) = 0 - (-x)|^ 3;
A5: (-x) to_power 2 = (-x)^2 by POWER:46
      .= x^2;
    (-x)|^ 3 = (-x) to_power (2+1) by POWER:41
      .= ((-x) to_power 2)*((-x) to_power 1) by A3,POWER:27;
    then
A6: (-x)|^ 3 = ((-x) to_power 2)*(-x);
    (-x)|^ 4 = x|^ 4 by Lm1,POWER:1;
    then x|^ 4 = (-x) to_power (3+1) by POWER:41
      .= ((-x) to_power 3)*((-x) to_power 1) by A3,POWER:27
      .= ((-x)|^ 3)*((-x) to_power 1);
    then x|^ 4 = ((-x)|^ 3)*(-x)
      .= x^2*(x*x) by A6,A5;
    hence thesis by A6,A5,A4;
  end;
end;
