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|^(2*m+3) + b1|^(2*m+3) = (a1|^(2*m+2)+b1|^(2*m+2))*(a1+b1)
  - a1*b1*(a1|^(2*m+1)+b1|^(2*m+1))
  proof
    (a1|^(2*m+2)+b1|^(2*m+2))*(a1+b1) - a1*b1*(a1|^(2*m+1)+b1|^(2*m+1))
    = a1*(a1|^(2*m+2)+b1|^(2*m+2))+b1*(a1|^(2*m+2)+b1|^(2*m+2))
    - a1*b1*a1|^(2*m+1)-a1*(b1*b1|^(2*m+1))
    .= a1*(a1|^(2*m+2)+b1|^(2*m+2))+b1*(a1|^(2*m+2)+b1|^(2*m+2))
    - b1*(a1*a1|^(2*m+1))-a1*b1|^(2*m+1+1) by NEWTON:6
    .= a1*(a1|^(2*m+2)+b1|^(2*m+2))+b1*(a1|^(2*m+2)+b1|^(2*m+2))
    - b1*a1|^(2*m+1+1)-a1*b1|^(2*m+2) by NEWTON:6
    .= a1*a1|^(2*m+2)+b1*b1|^(2*m+2)
    .= a1|^(2*m+2+1)+b1*b1|^(2*m+2) by NEWTON:6
    .=a1|^(2*m+3)+ b1|^(2*m+2+1) by NEWTON:6;
    hence thesis;
  end;
