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

theorem Th2:
  for p be Real st 1 <= p for a,b be Real_Sequence
   for n be Nat
holds (Partial_Sums((a + b) rto_power p).n) to_power (1/p) <= (
Partial_Sums(a rto_power p).n) to_power (1/p) + (Partial_Sums(b rto_power p).n)
  to_power (1/p)
proof
  let p be Real such that
A1: 1<=p;
   reconsider p as Real;
  let a,b be Real_Sequence;
  set ap = a rto_power p;
  set bp = b rto_power p;
  set ab= ((a + b) rto_power p);
  let n be Nat;
  now
    per cases by A1,XXREAL_0:1;
    case
A2:   p > 1;
      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;
      hence thesis by A2,HOLDER_1:7;
    end;
    case
      p=1;
      hence thesis by Lm3;
    end;
  end;
  hence thesis;
end;
