reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

theorem
  (a-b)|^3 = a|^3-3*a|^2*b+3*b|^2*a-b|^3
proof
  (a-b)|^(2+1)=((a-b)|^2)*(a-b) by NEWTON:6
    .=(a|^2-2*a*b+b|^2)*(a-b) by Lm5
    .=a|^2*a-a|^2*b-(2*a*b*a-2*a*b*b)+(b|^2*a-b|^2*b)
    .=a|^2*a-a|^2*b-(2*a*b*a-2*a*b*b)+(b|^2*a-b|^(2+1)) by NEWTON:6
    .=a|^3-a|^2*b-(2*(a*a)*b-2*a*b*b)+(b|^2*a-b|^3) by NEWTON:6
    .=a|^3-a|^2*b-(2*(a|^1*a)*b-2*a*b*b)+(b|^2*a-b|^3)
    .=a|^3-a|^2*b-(2*(a|^(1+1))*b-2*a*b*b)+(b|^2*a-b|^3) by NEWTON:6
    .=a|^3-3*a|^2*b+2*(b*b)*a+b|^2*a-b|^3
    .=a|^3-3*a|^2*b+2*(b|^1*b)*a+b|^2*a-b|^3
    .=a|^3-3*a|^2*b+2*b|^(1+1)*a+b|^2*a-b|^3 by NEWTON:6
    .=a|^3-3*a|^2*b+3*b|^2*a-b|^3;
  hence thesis;
end;
