reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for a,b be non negative Real, n be non zero Nat holds
    a*(a+b)|^n + (a+b)*b|^n <= (a+b)|^(n+1)
  proof
    let a,b be non negative Real, n be non zero Nat;
    a*((a+b)|^n + b|^n) +  b*b|^n <= ((a+b)|^(n+1) - b|^(n+1)) + b*b|^n
      by PLT,XREAL_1:6; then
    a*(a+b)|^n + (a+b)*b|^n <= (a+b)|^(n+1)- b|^(n+1) + b|^(n+1) by NEWTON:6;
    hence thesis;
  end;
