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 - b1|^m = (a1-b1)*k implies
  a1|^(m+2) - b1|^(m+2) = (a1|^(m+1)+b1|^(m+1) + a1*b1*k)*(a1-b1)
  proof
    assume a1|^m - b1|^m = (a1-b1)*k;
    hence a1|^(m+2) - b1|^(m+2) =
    (a1|^(m+1)+b1|^(m+1))*(a1-b1) + a1*b1*((a1-b1)*k) by Th21
    .= (a1|^(m+1)+b1|^(m+1) + a1*b1*k)*(a1-b1);
  end;
