reserve a, b, p, q for Real;

theorem
  for p be Real st 1 < p for a,b,ap,bp,ab be Real_Sequence st ( for n be
Nat holds ap.n=|.a.n.| to_power p & bp.n=|.b.n.| to_power p & ab.n
  =|.a.n+b.n.| to_power p ) & ap is summable & bp is summable holds ab is
summable & Sum(ab) to_power (1/p) <= Sum(ap) to_power (1/p) + Sum(bp) to_power
  (1/p)
proof
  let p be Real such that
A1: 1 < p;
A2: 1/p > 0 by A1,XREAL_1:139;
  let a,b,ap,bp,ab be Real_Sequence such that
A3: for n be Nat holds ap.n=|.a.n.| to_power p & bp.n=|.b.
  n.| to_power p & ab.n=|.a.n+ b.n.| to_power p and
A4: ap is summable and
A5: bp is summable;
  deffunc H(Nat)=(Partial_Sums(ab).$1) to_power (1/p);
  consider z be Real_Sequence such that
A6: for n be Nat holds z.n=H(n) from SEQ_1:sch 1;
A7: for n be Nat holds 0 <= ab.n
  proof
    let n be Nat;
A8: ab.n=|.a.n +b.n.| to_power p by A3;
    per cases by COMPLEX1:46;
    suppose
      |.a.n+b.n.| = 0;
      hence thesis by A1,A8,POWER:def 2;
    end;
    suppose
      |.a.n+b.n.| > 0;
      hence thesis by A8,POWER:34;
    end;
  end;
A9: for n be Nat holds 0 <= z.n
  proof
    let n be Nat;
A10: z.n=(Partial_Sums(ab).n) to_power (1/p) by A6;
    per cases by A7,Lm2;
    suppose
      Partial_Sums(ab).n = 0;
      hence thesis by A2,A10,POWER:def 2;
    end;
    suppose
      Partial_Sums(ab).n > 0;
      hence thesis by A10,POWER:34;
    end;
  end;
A11: Partial_Sums(ab) is non-decreasing by A7,SERIES_1:16;
A12: now
    let n,m be Nat;
    assume n <= m;
    then
A13: Partial_Sums(ab).n <= Partial_Sums(ab).m by A11,SEQM_3:6;
A14: 0 <= (Partial_Sums(ab).m) to_power (1/p)
    proof
      per cases by A7,Lm2;
      suppose
        Partial_Sums(ab).m = 0;
        hence thesis by A2,POWER:def 2;
      end;
      suppose
        Partial_Sums(ab).m > 0;
        hence thesis by POWER:34;
      end;
    end;
    now
      per cases;
      case
        Partial_Sums(ab).n = Partial_Sums(ab).m;
        hence (Partial_Sums(ab).n) to_power (1/p) <= (Partial_Sums(ab).m)
        to_power (1/p);
      end;
      case
        Partial_Sums(ab).n <> Partial_Sums(ab).m;
        then
A15:    Partial_Sums(ab).n < Partial_Sums(ab).m by A13,XXREAL_0:1;
        now
          per cases;
          case
            Partial_Sums(ab).n =0;
            hence (Partial_Sums(ab).n) to_power (1/p) <= (Partial_Sums(ab).m)
            to_power (1/p) by A2,A14,POWER:def 2;
          end;
          case
A16:        Partial_Sums(ab).n <> 0;
            0 <= Partial_Sums(ab).n by A7,Lm2;
            hence (Partial_Sums(ab).n) to_power (1/p) <= (Partial_Sums(ab).m)
            to_power (1/p) by A2,A15,A16,POWER:37;
          end;
        end;
        hence (Partial_Sums(ab).n) to_power (1/p) <= (Partial_Sums(ab).m)
        to_power (1/p);
      end;
    end;
    hence (Partial_Sums(ab).n) to_power (1/p) <= (Partial_Sums(ab).m) to_power
    (1/p);
  end;
  now
    let n,m be Nat;
    assume
A17: n <= m;
    z.n =(Partial_Sums(ab).n) to_power (1/p) & z.m =(Partial_Sums(ab).m)
    to_power (1/p) by A6;
    hence z.n <= z.m by A12,A17;
  end;
  then
A18: z is non-decreasing by SEQM_3:6;
  for n be Nat holds 0 <= ap.n
  proof
    let n be Nat;
A19: ap.n=|.a.n.| to_power p by A3;
    now
      per cases by COMPLEX1:46;
      case
        |.a.n.| = 0;
        hence thesis by A1,A19,POWER:def 2;
      end;
      case
        |.a.n.| > 0;
        hence thesis by A19,POWER:34;
      end;
    end;
    hence thesis;
  end;
  then
A20: 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
A21: for n be Nat holds x.n=F(n) from SEQ_1:sch 1;
  for n be Nat holds 0 <= bp.n
  proof
    let n be Nat;
A22: bp.n=|.b.n.| to_power p by A3;
    now
      per cases by COMPLEX1:46;
      case
        |.b.n.| = 0;
        hence thesis by A1,A22,POWER:def 2;
      end;
      case
        |.b.n.| > 0;
        hence thesis by A22,POWER:34;
      end;
    end;
    hence thesis;
  end;
  then
A23: for n be Nat holds 0 <= Partial_Sums(bp).n by Lm2;
  deffunc G(Nat)=(Partial_Sums(bp).$1) to_power (1/p);
  consider y be Real_Sequence such that
A24: for n be Nat holds y.n=G(n) from SEQ_1:sch 1;
A25: Partial_Sums(bp) is convergent by A5,SERIES_1:def 2;
  then
A26: y is convergent by A2,A24,A23,Th10;
A27: Partial_Sums(ap) is convergent by A4,SERIES_1:def 2;
  then
A28: x is convergent by A2,A21,A20,Th10;
A29: for n be Nat holds z.n <=x.n + y.n
  proof
    let n be Nat;
A30: y.n = (Partial_Sums(bp).n) to_power (1/p) by A24;
    (Partial_Sums(ab).n) to_power (1/p) <= ( (Partial_Sums(ap).n)
    to_power (1/p) ) + ( (Partial_Sums(bp).n) to_power (1/p) ) & x.n = (
    Partial_Sums(ap).n) to_power (1/p) by A1,A3,A21,Th7;
    hence thesis by A6,A30;
  end;
  then
A31: z is convergent by A28,A26,A18,Th9;
A32: for n be Nat holds (z.n) to_power p = Partial_Sums(ab).n
  proof
    let n be Nat;
A33: z.n = (Partial_Sums(ab).n) to_power (1/p) by A6;
    now
      per cases;
      case
A34:    Partial_Sums(ab).n =0;
        then z.n= 0 by A2,A33,POWER:def 2;
        hence thesis by A1,A34,POWER:def 2;
      end;
      case
A35:    Partial_Sums(ab).n <> 0;
        0 <= Partial_Sums(ab).n by A7,Lm2;
        hence (z.n) to_power p = ( Partial_Sums(ab).n) to_power ((1/p)*p) by
A33,A35,POWER:33
          .=( Partial_Sums(ab).n) to_power 1 by A1,XCMPLX_1:106
          .= Partial_Sums(ab).n by POWER:25;
      end;
    end;
    hence thesis;
  end;
  then lim Partial_Sums(ab) = (lim z) to_power p by A1,A9,A31,Th10;
  then
A36: Sum(ab)= (lim z) to_power p by SERIES_1:def 3;
A37: Sum(ab) to_power (1/p) = lim z
  proof
    per cases;
    suppose
A38:  lim z=0;
      hence Sum(ab) to_power (1/p) = 0 to_power (1/p) by A1,A36,POWER:def 2
        .= lim z by A2,A38,POWER:def 2;
    end;
    suppose
      lim z <> 0;
      then 0 < lim z by A9,A31,SEQ_2:17;
      hence (Sum ab) to_power (1/p) = (lim z) to_power ((1/p)*p) by A36,
POWER:33
        .= (lim z) to_power 1 by A1,XCMPLX_1:106
        .= lim z by POWER:25;
    end;
  end;
  Sum(bp) = lim Partial_Sums(bp) by SERIES_1:def 3;
  then
A39: lim y= Sum(bp) to_power (1/p) by A2,A24,A25,A23,Th10;
  Sum(ap) = lim Partial_Sums(ap) by SERIES_1:def 3;
  then
A40: lim x= Sum(ap) to_power (1/p) by A2,A21,A27,A20,Th10;
  Partial_Sums(ab) is convergent by A1,A9,A31,A32,Th10;
  hence thesis by A28,A40,A26,A39,A29,A18,A37,Th9,SERIES_1:def 2;
end;
