reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem N3:
  a1|^3 - b1|^3 = (a1 - b1)*(a1|^2 + b1|^2 + a1*b1)
  proof
    a1|^(2+1) - b1|^(2+1) = a1|^2*(a1|^1 - b1|^1) + b1|^1*(a1|^2 - b1|^2)
      by RI2
    .= a1|^2*(a1-b1)+b1*((a1+b1)*(a1-b1)) by NEWTON01:1
    .= (a1-b1)*(a1|^2+ b1*b1 + a1*b1 );
    hence thesis by NEWTON:81;
  end;
