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

theorem SLT:
  for a,b be non negative Real, n be non zero Nat holds
    a|^n + b|^n <= Sum((a,b) Subnomial n)
  proof
    let a,b be non negative Real, n be non zero Nat;
    reconsider f = (a,b) Subnomial n
      as nonnegative-yielding FinSequence of REAL;
    len f = n+1 by Def2; then
    A1: Sum f = Sum((f|n)/^1) + f.1 + f.(n+1) by NEWTON02:115;
    A2: ((a,b) Subnomial n).1 = ((a,b) In_Power n).1 by NS
    .= a|^n by NEWTON:28;
    A3: ((a,b) Subnomial n).(n+1) = ((a,b) In_Power n).(n+1) by NS
    .= b|^n by NEWTON:29;
     Sum((f|n)/^1) + (a|^n + b|^n) >= 0 + (a|^n + b|^n) by XREAL_1:6;
    hence thesis by A1,A2,A3;
  end;
