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

theorem
  a1|^(m+2) + b1|^(m+2) = (a1|^(m+1)-b1|^(m+1))*(a1-b1) + a1*b1*(a1|^m+b1|^m)
  proof
    (a1|^(m+1)-b1|^(m+1))*(a1-b1) + a1*b1*(a1|^m+b1|^m)
    = a1|^(m+1)*a1-b1|^(m+1)*a1 -
    a1|^(m+1)*b1 + b1|^(m+1)*b1 + a1*b1*(a1|^m+b1|^m)
    .=a1|^(m+1+1) - b1|^(m+1)*a1 - a1|^(m+1)*b1 + b1|^(m+1)*b1
    + a1*b1*(a1|^m + b1|^m) by NEWTON:6
    .=a1|^(m+2) - b1|^(m+1)*a1 - a1|^(m+1)*b1 + b1|^(m+1+1)
    + a1*b1*(a1|^m + b1|^m) by NEWTON:6
    .=a1|^(m+2) - b1|^(m+1)*a1 - a1|^(m+1)*b1 + b1|^(m+2) +
    b1*(a1*a1|^m) + a1*b1*b1|^m
    .=a1|^(m+2)-b1|^(m+1)*a1-a1|^(m+1)*b1 + b1|^(m+2)
    + b1*a1|^(m+1)+a1*(b1*b1|^m) by NEWTON:6
    .=a1|^(m+2)- b1|^(m+1)*a1 + b1|^(m+2) + a1*b1|^(m+1) by NEWTON:6;
    hence thesis;
  end;
