reserve a, b, p, q for Real;

theorem
  for p, q be Real st 1 < p & 1/p + 1/q = 1 for a,b,ap,bq,ab be
Real_Sequence st ( for n be Nat holds ap.n=|.a.n.| to_power p & bq.
n=|.b.n.| to_power q & ab.n=|.a.n* b.n.| ) & ap is summable & bq is summable
  holds ab is summable & Sum ab <= ( Sum(ap) to_power (1/p) ) * ( Sum(bq)
  to_power (1/q) )
proof
  let p,q be Real such that
A1: 1 < p and
A2: 1/p + 1/q = 1;
  let a,b,ap,bq,ab be Real_Sequence such that
A3: for n be Nat holds ap.n=|.a.n.| to_power p & bq.n=|.b.
  n.| to_power q & ab.n=|.a.n* b.n.| and
A4: ap is summable and
A5: bq is summable;
A6: Partial_Sums(ap) is convergent by A4,SERIES_1:def 2;
  reconsider pp=1/p as Real;
  1/p < 1 by A1,XREAL_1:189;
  then
A7: 1-1 < 1-pp by XREAL_1:15;
  then
A8: 0 < q by A2;
  for n be Nat holds 0 <= bq.n
  proof
    let n be Nat;
A9: bq.n=|.b.n.| to_power q by A3;
    now
      per cases by COMPLEX1:46;
      case
        |.b.n.| = 0;
        hence thesis by A8,A9,POWER:def 2;
      end;
      case
        |.b.n.| > 0;
        hence thesis by A9,POWER:34;
      end;
    end;
    hence thesis;
  end;
  then
A10: for n be Nat holds 0 <= Partial_Sums(bq).n by Lm2;
  for n be Nat holds 0 <= ab.n
  proof
    let n be Nat;
    ab.n= |.a.n*b.n.| by A3;
    hence thesis by COMPLEX1:46;
  end;
  then
A11: Partial_Sums(ab) is non-decreasing by SERIES_1:16;
  deffunc G(Nat)=(Partial_Sums(bq).$1) to_power (1/q);
  consider y be Real_Sequence such that
A12: for n be Nat holds y.n=G(n) from SEQ_1:sch 1;
  for n be Nat holds 0 <= ap.n
  proof
    let n be Nat;
A13: ap.n=|.a.n.| to_power p by A3;
    now
      per cases by COMPLEX1:46;
      case
        |.a.n.| = 0;
        hence thesis by A1,A13,POWER:def 2;
      end;
      case
        |.a.n.| > 0;
        hence thesis by A13,POWER:34;
      end;
    end;
    hence thesis;
  end;
  then
A14: for n be Nat holds 0 <= Partial_Sums(ap).n by Lm2;
  deffunc F(Nat)=(Partial_Sums(ap).$1) to_power (1/p);
  consider x be Real_Sequence such that
A15: for n be Nat holds x.n=F(n) from SEQ_1:sch 1;
A16: for n be Nat holds Partial_Sums(ab).n <=(x(#)y).n
  proof
    let n be Nat;
A17: y.n = (Partial_Sums(bq).n) to_power (1/q) by A12;
    Partial_Sums(ab).n <= ( (Partial_Sums(ap).n) to_power (1/p) ) * ( (
Partial_Sums(bq).n) to_power (1/q) ) & x.n = (Partial_Sums(ap).n) to_power (1/
    p) by A1,A2,A3,A15,Th6;
    hence thesis by A17,SEQ_1:8;
  end;
A18: 1/p > 0 by A1,XREAL_1:139;
  then
A19: x is convergent by A15,A6,A14,Th10;
A20: Partial_Sums(bq) is convergent by A5,SERIES_1:def 2;
  then
A21: y is convergent by A2,A7,A12,A10,Th10;
  then x(#)y is convergent by A19;
  then
A22: Partial_Sums(ab) is convergent & lim Partial_Sums(ab) <= lim (x(#)y) by
A16,A11,Th8;
  Sum ap = lim Partial_Sums(ap) by SERIES_1:def 3;
  then
A23: lim x= Sum(ap) to_power (1/p) by A18,A15,A6,A14,Th10;
  Sum(bq) = lim Partial_Sums(bq) by SERIES_1:def 3;
  then lim y= Sum(bq) to_power (1/q) by A2,A7,A12,A20,A10,Th10;
  then
  lim (x(#)y) = (Sum(ap) to_power (1/p))* (Sum(bq) to_power (1/q)) by A19,A23
,A21,SEQ_2:15;
  hence thesis by A22,SERIES_1:def 2,def 3;
end;
