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
  a|^(2*m+2)-b|^(2*m+2) =
  (a|^2-b|^2)*(c*(a|^2+b|^2)+a|^(2*m)+b|^(2*m))/2
  iff a|^(2*m) - b|^(2*m) = (a|^2-b|^2)*c
  proof
    set k = a|^2; set l = b|^2;
    A1: a|^(2*m)=(a|^2)|^m & b|^(2*m)=(b|^2)|^m by NEWTON:9;
    A2: a|^(2*m+2) = a|^(2*(m+1))
    .=(a|^2)|^(m+1) by NEWTON:9;
    b|^(2*m+2) = b|^(2*(m+1))
    .=(b|^2)|^(m+1) by NEWTON:9;
    hence thesis by A1,A2,Th14;
  end;
