theorem :: NAT_5:1 extended
  for a be non trivial Nat holds
    a|^(n+1) + a|^n < a|^(n+2)
  proof
    let a be non trivial Nat;
A0: not (a = 0 or a = 1) by NAT_2:def 1; then
A0a: a >= 2 by NAT_1:23;
A1: a|^(n+1+1) = a*a|^(n+1) by NEWTON:6;
A2: a*a|^(n+1) >= 2*a|^(n+1) by A0,NAT_1:23,XREAL_1:64;
    a > 1 by A0a,XXREAL_0:2; then
    a*a|^n > 1*a|^n by XREAL_1:68; then
    a|^(n+1) > a|^n by NEWTON:6; then
    a|^(n+1) + a|^(n+1) > a|^(n+1) + a|^n by XREAL_1:6;
    hence thesis by A1,A2,XXREAL_0:2;
  end;
