reserve x1,x2,y1,a,b,c for Real;

theorem Th3:
  for a,b be Real_Sequence for p be Real st 1 <= p & (a rto_power p
  ) is summable & (b rto_power p) is summable holds ((a+b) rto_power p) is
  summable & (Sum((a + b) rto_power p)) to_power (1/p) <= (Sum(a rto_power p))
  to_power (1/p) + (Sum(b rto_power p)) to_power (1/p)
proof
  let a,b be Real_Sequence;
  let p be Real such that
A1: 1<=p and
A2: (a rto_power p) is summable and
A3: (b rto_power p) is summable;
  set ab= ((a + b) rto_power p);
  set bp = b rto_power p;
  set ap = a rto_power p;
A4: now
    let n be Nat;
    thus ap.n= |.a .n.| to_power p by Def1;
    thus bp.n= |.b .n.| to_power p by Def1;
    ((a + b) rto_power p).n =|.(a + b).n.| to_power p by Def1
      .=|.a.n+b.n.| to_power p by SEQ_1:7;
    hence ab.n= |.a.n+b.n.| to_power p;
  end;
  reconsider p as Real;
  now
    per cases by A1,XXREAL_0:1;
    case
      p > 1;
      hence Sum(ab) to_power (1/p) <= Sum ap to_power (1/p) + Sum(bp) to_power
      (1/p) & ab is summable by A2,A3,A4,HOLDER_1:13;
    end;
    case
      p=1;
      hence
      Sum(ab) to_power (1/p) <= Sum(ap) to_power (1/p) + Sum(bp) to_power
      (1/p) & ab is summable by A2,A3,Lm4;
    end;
  end;
  hence thesis;
end;
