reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem Th70:
for k be geq_than_1 Real, Sq be sequence of Lp-Space(M,k) st
  Sq is Cauchy_sequence_by_Norm holds Sq is convergent
proof
    let k be geq_than_1 Real;
    let Sq be sequence of Lp-Space(M,k);
A1:1 <= k by Def1;
    assume A2: Sq is Cauchy_sequence_by_Norm;
    consider Fsq be with_the_same_dom Functional_Sequence of X,REAL such that
A3:  for n be Nat holds Fsq.n in Lp_Functions(M,k) & Fsq.n in Sq.n &
      Sq.n= a.e-eq-class_Lp(Fsq.n,M,k) &
      ex r be Real st 0<=r & r = Integral(M,(abs (Fsq.n)) to_power k) &
        ||. Sq.n .|| =r to_power (1/k) by Th63;
    Fsq.0 in Lp_Functions(M,k) by A3; then
A4: ex D be Element of S st M.D` =0 & dom(Fsq.0) = D &
     (Fsq.0) is D-measurable by Th35; then
    reconsider E = dom(Fsq.0) as Element of S;
    consider N be increasing sequence of NAT such that
A5:  for i,j be Nat st j >= N.i holds
      ||. Sq.j - Sq.(N.i) .|| < 2 to_power (-i) by Th65,A2;
    deffunc FsqN(Nat) = Fsq.(N.$1);
    consider F1 be Functional_Sequence of X,REAL such that
A6:  for n be Nat holds F1.n = FsqN(n) from SEQFUNC:sch 1;
A7: for n be Nat holds dom(F1.n) = E & F1.n in Lp_Functions(M,k) &
      F1.n is E-measurable & abs(F1.n) in Lp_Functions(M,k)
    proof
     let n be Nat;
A8: F1.n = Fsq.(N.n) by A6;
     hence
A9:   dom(F1.n) = E & F1.n in Lp_Functions(M,k) by A3,MESFUNC8:def 2; then
     consider F be PartFunc of X,REAL such that
Z1:  F1.n = F & 
     ex ND be Element of S st M.ND` =0 & dom F = ND &
      F is ND-measurable & (abs F) to_power k is_integrable_on M;
     consider ND be Element of S such that 
Z2:   M.ND` =0 & dom F = ND &
      F is ND-measurable & (abs F) to_power k is_integrable_on M by Z1;
     ND = E by Z1,Z2,A8,MESFUNC8:def 2; 
     hence F1.n is E-measurable by Z1,Z2;
     thus abs(F1.n) in Lp_Functions(M,k) by A9,Th28;
    end;
    for n,m be Nat holds dom (F1.n) = dom (F1.m)
    proof
     let n,m be Nat;
     dom(F1.n) = E & dom(F1.m) = E by A7;
     hence thesis;
    end; then
    reconsider F1 as with_the_same_dom Functional_Sequence of X,REAL
      by MESFUNC8:def 2;
    deffunc FF(Nat) = F1.($1+1) - F1.$1;
    consider FMF be Functional_Sequence of X,REAL such that
A10:  for n be Nat holds FMF.n = FF(n) from SEQFUNC:sch 1;
A11: for n be Nat holds dom(FMF.n) = E & FMF.n in Lp_Functions(M,k)
    proof
     let n be Nat;
A12:  dom(F1.n) = E & dom(F1.(n+1)) = E by A7;
     FMF.n = F1.(n+1) - F1.n by A10; then
     dom(FMF.n) = dom (F1.(n+1)) /\ dom (F1.n) by VALUED_1:12;
     hence dom(FMF.n) = E by A12;
     Fsq.(N.(n+1)) in Lp_Functions(M,k) & Fsq.(N.n) in Lp_Functions(M,k)
       by A3; then
     F1.(n+1) in Lp_Functions(M,k) & F1.n in Lp_Functions(M,k) by A6; then
     F1.(n+1) - F1.n in Lp_Functions(M,k) by Th27;
     hence FMF.n in Lp_Functions(M,k) by A10;
    end;
    for n,m be Nat holds dom(FMF.n) = dom(FMF.m)
    proof
     let n,m be Nat;
     dom(FMF.n) = E & dom(FMF.m) = E by A11;
     hence thesis;
    end; then
    reconsider FMF as with_the_same_dom Functional_Sequence of X,REAL
       by MESFUNC8:def 2;
    set AbsFMF = abs FMF;
A13: for n be Nat holds
     AbsFMF.n is nonnegative & dom(AbsFMF.n) = E & abs(AbsFMF.n) = AbsFMF.n &
     AbsFMF.n in Lp_Functions(M,k) & AbsFMF.n is E-measurable
    proof
     let n be Nat;
A14:  AbsFMF.n = abs (FMF.n) by SEQFUNC:def 4;
     hence AbsFMF.n is nonnegative;
A15:  dom(FMF.n) = E & FMF.n in Lp_Functions(M,k) by A11;
     hence dom(AbsFMF.n) = E & abs(AbsFMF.n) = AbsFMF.n by A14,VALUED_1:def 11;
     thus AbsFMF.n in Lp_Functions(M,k) by A11,A14,Th28; then
     consider D be Element of S such that 
Z1:  M.D` = 0 & dom(AbsFMF.n) = D &
       AbsFMF.n is D-measurable by Th35;
     D = E by Z1,A15,A14,VALUED_1:def 11; 
     hence AbsFMF.n is E-measurable by Z1;
    end;
    reconsider
      AbsFMF as with_the_same_dom Functional_Sequence of X,REAL by Th69;
    deffunc Gk(Nat) =abs(F1.0) + (Partial_Sums AbsFMF).$1;
    consider G be Functional_Sequence of X,REAL such that
A16: for n be Nat holds G.n = Gk(n) from SEQFUNC:sch 1;
A17:for n be Nat holds
      dom(G.n) = E & G.n in Lp_Functions(M,k) & G.n is nonnegative &
      G.n is E-measurable & abs(G.n) = G.n
    proof
     let n be Nat;
A18:  G.n = abs(F1.0) + (Partial_Sums AbsFMF).n by A16; then
A19:  dom(G.n)
        = dom(abs(F1.0)) /\ dom((Partial_Sums AbsFMF).n) by VALUED_1:def 1
       .= dom(F1.0) /\ dom((Partial_Sums AbsFMF).n) by VALUED_1:def 11
       .= dom(F1.0) /\ dom(AbsFMF.0) by MESFUN9C:11;
A20:  (Partial_Sums AbsFMF).n in Lp_Functions(M,k) &
     (Partial_Sums AbsFMF).n is nonnegative &
     (Partial_Sums AbsFMF).n is E-measurable
        by A13,Th66,Th67,MESFUN9C:16;
A21:  dom(AbsFMF.0) = E by A13;
A22:  F1.0 in Lp_Functions(M,k) & dom(F1.0) = E & F1.0 is E-measurable
        by A7; then
     abs(F1.0) in Lp_Functions(M,k) & abs(F1.0) is nonnegative &
     abs(F1.0) is E-measurable by Th28,MESFUNC6:48;
     hence thesis by A19,A22,A21,A18,A20,Th14,Th25,MESFUNC6:26,56;
    end;
    deffunc Gpk(Nat) =(G.$1) to_power k;
    consider Gp be Functional_Sequence of X,REAL such that
A23: for n be Nat holds Gp.n = Gpk(n) from SEQFUNC:sch 1;
A24:for n be Nat holds
     ((G.n) to_power k) is nonnegative & ((G.n) to_power k) is E-measurable
    proof
     let n be Nat;
A25:  G.n is nonnegative by A17;
     hence ((G.n) to_power k) is nonnegative;
     G.n is E-measurable & dom(G.n) = E by A17;
     hence ((G.n) to_power k) is E-measurable by A25,MESFUN6C:29;
    end;
    reconsider ExtGp = R_EAL Gp as Functional_Sequence of X,ExtREAL;
A26:for n be Nat holds
      dom(ExtGp.n) = E & ExtGp.n is E-measurable &
      ExtGp.n is nonnegative
    proof
     let n be Nat;
     ExtGp.n = R_EAL((G.n) to_power k) by A23; then
     dom(ExtGp.n) = dom (G.n) by MESFUN6C:def 4;
     hence dom(ExtGp.n) = E by A17;
     ((G.n) to_power k) is E-measurable by A24; then
     R_EAL((G.n) to_power k) is E-measurable;
     hence ExtGp.n is E-measurable by A23;
     ((G.n) to_power k) is nonnegative by A24;
     hence ExtGp.n is nonnegative by A23;
    end; then
A27:dom(ExtGp.0) = E & ExtGp.0 is nonnegative;
    for n,m be Nat holds dom (ExtGp.n) = dom (ExtGp.m)
    proof
     let n,m be Nat;
     dom (ExtGp.n) = E & dom(ExtGp.m) = E by A26;
     hence thesis;
    end; then
    reconsider ExtGp as with_the_same_dom Functional_Sequence of X,ExtREAL
      by MESFUNC8:def 2;
A28:for n,m be Nat st n <=m holds
     for x be Element of X st x in E holds (ExtGp.n).x <= (ExtGp.m).x
    proof
     let n,m be Nat;
     assume A29:n <=m;
     let x be Element of X;
     assume A30:x in E; then
A31:  x in dom (G.n) & x in dom (G.m) by A17; then
     x in dom((G.n) to_power k) &
     x in dom((G.m) to_power k) by MESFUN6C:def 4; then
     ((G.n).x) to_power k = ((G.n) to_power k).x &
     ((G.m).x) to_power k = ((G.m) to_power k).x by MESFUN6C:def 4; then
A32:  ((G.n).x) to_power k = (ExtGp.n).x &
     ((G.m).x) to_power k = (ExtGp.m).x by A23;
     dom(AbsFMF.0) = E by A13; then
     ((Partial_Sums AbsFMF).n).x <= ((Partial_Sums AbsFMF).m).x
        by Th68,A29,A30,A13; then
A33:  (abs(F1.0)).x + ((Partial_Sums AbsFMF).n).x
       <= (abs(F1.0)).x + ((Partial_Sums AbsFMF).m).x by XREAL_1:6;
     G.m= abs(F1.0) + (Partial_Sums AbsFMF).m &
     G.n= abs(F1.0) + (Partial_Sums AbsFMF).n by A16; then
A34:  (G.m).x= (abs(F1.0)).x + ((Partial_Sums AbsFMF).m).x &
     (G.n).x= (abs(F1.0)).x + ((Partial_Sums AbsFMF).n).x by A31,VALUED_1:def 1
;
     G.n is nonnegative by A17; then
     0 <= (G.n).x by MESFUNC6:51;
     hence thesis by A32,A33,A34,HOLDER_1:3;
    end;
A35:for x be Element of X st x in E holds ExtGp#x is non-decreasing
    proof
     let x be Element of X;
     assume A36: x in E;
     for n,m be Nat st m<=n holds ((ExtGp)#x).m <= ((ExtGp)#x).n
     proof
      let n,m be Nat;
      assume m <= n; then
      ((ExtGp).m).x <= ((ExtGp).n).x by A28,A36; then
      ((ExtGp)#x).m <= ((ExtGp).n).x by MESFUNC5:def 13;
      hence thesis by MESFUNC5:def 13;
     end;
     hence ExtGp#x is non-decreasing by RINFSUP2:7;
    end;
A37:for x be Element of X st x in E holds ExtGp#x is convergent
    proof
     let x be Element of X;
     assume x in E; then
     ExtGp#x is non-decreasing by A35;
     hence thesis by RINFSUP2:37;
    end; then
    consider I be ExtREAL_sequence such that
A38: (for n be Nat holds I.n = Integral(M,ExtGp.n)) &
     I is convergent & Integral(M,lim ExtGp) = lim I
        by A27,A26,A28,MESFUNC9:52;
    now let y be object;
     assume y in rng I; then
     consider x be Element of NAT such that
A39:   y = I.x by FUNCT_2:113;
A40:  y = Integral(M,Gp.x) by A39,A38;
     G.x = abs(G.x) by A17; then
A41:  Gp.x = abs(G.x) to_power k by A23;
     G.x in Lp_Functions(M,k) by A17;
     hence y in REAL by A40,A41,Th49;
    end; then
    rng I c= REAL; then
    reconsider Ir = I as sequence of REAL by FUNCT_2:6;
    deffunc KAbsFMF(Nat) = Integral(M,(AbsFMF.$1) to_power k);
A42:for x being Element of NAT holds KAbsFMF(x) is Element of REAL
    proof
     let x being Element of NAT;
     (AbsFMF.x) in Lp_Functions(M,k) by A13; then
     Integral(M,(abs(AbsFMF.x)) to_power k) in REAL by Th49;
     hence thesis by A13;
    end;
    consider KPAbsFMF being sequence of REAL such that
A43: for x be Element of NAT holds
      KPAbsFMF.x = KAbsFMF(x) from FUNCT_2:sch 9(A42);
    deffunc KKAbsFMF(Nat) = (KPAbsFMF.$1) to_power (1/k);
A44:for x being Element of NAT holds KKAbsFMF(x) is Element of REAL
            by XREAL_0:def 1;
    consider PAbsFMF being sequence of REAL such that
A45: for x be Element of NAT holds
       PAbsFMF.x = KKAbsFMF(x) from FUNCT_2:sch 9(A44);
    F1.0 in Lp_Functions(M,k) by A7; then
    reconsider
      RF0=Integral(M,(abs(F1.0)) to_power k) as Element of REAL by Th49;
    deffunc LAbsFMF(Nat)
      = RF0 to_power (1/k) + (Partial_Sums PAbsFMF).$1;
A46:for x being Element of NAT holds LAbsFMF(x) is Element of REAL
           by XREAL_0:def 1;
    consider QAbsFMF being sequence of REAL such that
A47: for x being Element of NAT holds QAbsFMF.x = LAbsFMF(x)
        from FUNCT_2:sch 9(A46);
A48:for n being Nat holds (Ir.n) to_power (1/k) <= QAbsFMF.n
    proof
     defpred PN[Nat] means (Ir.$1) to_power (1/k) <= QAbsFMF.$1;
A49:  abs(F1.0) in Lp_Functions(M,k) & AbsFMF.0 in Lp_Functions(M,k) by A13,A7;
     G.0 = abs(F1.0) + (Partial_Sums AbsFMF).0 by A16; then
A50:  G.0 =abs(F1.0) + AbsFMF.0 by MESFUN9C:def 2;
     Ir.0 = Integral(M,Gp.0) by A38; then
     Ir.0 = Integral(M,(G.0) to_power k) by A23; then
A51:  Ir.0 = Integral(M,abs( abs(F1.0) + AbsFMF.0 ) to_power k) by A17,A50;
     KPAbsFMF.0 = Integral(M,(AbsFMF.0) to_power k) by A43; then
A52:  KPAbsFMF.0 = Integral(M,abs(AbsFMF.0) to_power k) by A13;
A53:  RF0 = Integral(M,abs(abs(F1.0)) to_power k);
     QAbsFMF.0 = RF0 to_power (1/k) + (Partial_Sums PAbsFMF).0 by A47; then
     QAbsFMF.0 = RF0 to_power (1/k) + PAbsFMF.0 by SERIES_1:def 1; then
     QAbsFMF.0 = RF0 to_power (1/k) + (KPAbsFMF.0) to_power (1/k) by A45; then
A54:  PN[ 0] by A1,A49,A51,A52,A53,Th61;
A55:  now let n being Nat;
A56:    n in NAT by ORDINAL1:def 12;
      assume PN[n]; then
A57:   (Ir.n) to_power (1/k) + PAbsFMF.(n+1) <= QAbsFMF.n + PAbsFMF.(n+1)
         by XREAL_1:6;
      G.(n+1) = abs(F1.0) + (Partial_Sums AbsFMF).(n+1) by A16
      .= abs(F1.0) + ((Partial_Sums AbsFMF).n + AbsFMF.(n+1)) by MESFUN9C:def 2
      .= abs(F1.0) +(Partial_Sums AbsFMF).n + AbsFMF.(n+1) by RFUNCT_1:8; then
A58:   G.(n+1) = G.n + AbsFMF.(n+1) by A16;
A59:   AbsFMF.(n+1) in Lp_Functions(M,k) & G.n in Lp_Functions(M,k) by A13,A17;
      KPAbsFMF.(n+1) = Integral(M,(AbsFMF.(n+1)) to_power k) by A43; then
A60:   KPAbsFMF.(n+1) = Integral(M,abs (AbsFMF.(n+1)) to_power k) by A13;
A61:   PAbsFMF.(n+1) = (KPAbsFMF.(n+1)) to_power (1/k) by A45;
      Ir.n = Integral(M,Gp.n) & Ir.(n+1) = Integral(M,Gp.(n+1)) by A38; then
      Ir.n = Integral(M,(G.n) to_power k) &
      Ir.(n+1) = Integral(M,(G.(n+1)) to_power k) by A23; then
      Ir.n = Integral(M,abs (G.n) to_power k) &
      Ir.(n+1) =Integral(M,abs (G.n + AbsFMF.(n+1)) to_power k) by A58,A17;
 then
      (Ir.(n+1)) to_power (1/k) <= (Ir.n) to_power (1/k) + PAbsFMF.(n+1)
         by A1,A59,A60,A61,Th61; then
A62:   (Ir.(n+1)) to_power (1/k) <= QAbsFMF.n + PAbsFMF.(n+1) by A57,XXREAL_0:2
;
      QAbsFMF.n + PAbsFMF.(n+1)
       = RF0 to_power (1/k) + (Partial_Sums PAbsFMF).n + PAbsFMF.(n+1)
            by A47,A56
      .= RF0 to_power (1/k) + ((Partial_Sums PAbsFMF).n + PAbsFMF.(n+1))
      .= RF0 to_power (1/k) + (Partial_Sums PAbsFMF).(n+1) by SERIES_1:def 1;
      hence PN[n+1] by A62,A47;
     end;
     for n be Nat holds PN[n] from NAT_1:sch 2(A54,A55);
     hence thesis;
    end;
A63:for n be Nat holds PAbsFMF.n = ||. Sq.(N.(n+1))-Sq.(N.n) .||
    proof
     let n be Nat;
A64:   n in NAT by ORDINAL1:def 12;
     set m = N.n;
     set m1 = N.(n+1);
A65:  F1.(n+1) = Fsq.(N.(n+1)) & F1.n = Fsq.(N.n) by A6;
     AbsFMF.n = abs (FMF.n) by SEQFUNC:def 4; then
A66:  AbsFMF.n = abs( Fsq.(N.(n+1)) - Fsq.(N.n) ) by A65,A10;
A67:  Fsq.(N.(n+1)) in Lp_Functions(M,k) & Fsq.(N.(n+1)) in Sq.(N.(n+1)) &
     Fsq.(N.n) in Lp_Functions(M,k) & Fsq.(N.n) in Sq.m by A3; then
     (-1)(#)(Fsq.m) in (-1)*(Sq.m) by Th54; then
     Fsq.m1 - Fsq.m in Sq.m1 + (-1)*(Sq.m) by Th54,A67; then
     Fsq.m1 - Fsq.m in Sq.m1 - Sq.m by RLVECT_1:16; then
A68:  ex r be Real
     st 0<=r & r = Integral(M,(abs (Fsq.m1 - Fsq.m)) to_power k) &
      ||. Sq.m1 - Sq.m .|| = r to_power (1/k) by Th53;
     PAbsFMF.n = (KPAbsFMF.n) to_power (1/k) by A45,A64;
     hence thesis by A68,A66,A43,A64;
    end;
    1/2 < 1; then
    |.1/2.| < 1 by ABSVALUE:def 1; then
A69:(1/2) GeoSeq is summable & Sum((1/2) GeoSeq) = 1/(1-(1/2)) by SERIES_1:24;
    for n be Nat holds 0<=PAbsFMF.n & PAbsFMF.n <= ((1/2) GeoSeq).n
    proof
     let n be Nat;
A70: PAbsFMF.n = ||. Sq.(N.(n+1)) - Sq.(N.n) .|| by A63;
     hence 0 <= PAbsFMF.n;
     ((1/2) GeoSeq).n = (1/2) |^n by PREPOWER:def 1
                     .= (1/2) to_power n by POWER:41; then
A71: ((1/2) GeoSeq).n = 2 to_power (-n) by POWER:32;
     N is Real_Sequence by FUNCT_2:7,NUMBERS:19; then
     N.n < N.(n+1) by SEQM_3:def 6;
     hence PAbsFMF.n <= ((1/2) GeoSeq).n by A5,A70,A71;
    end; then
    PAbsFMF is summable & Sum(PAbsFMF) <= Sum((1/2) GeoSeq)
      by A69,SERIES_1:20; then
    Partial_Sums(PAbsFMF) is convergent by SERIES_1:def 2; then
    Partial_Sums(PAbsFMF) is bounded; then
    consider Br be Real such that
A72: for n be Nat holds Partial_Sums(PAbsFMF).n < Br by SEQ_2:def 3;
    for n being Nat holds
      Ir.n < (RF0 to_power (1/k) + Br) to_power k
    proof
     let n being Nat;
A73:   n in NAT by ORDINAL1:def 12;
     (Ir.n) to_power (1/k) <= QAbsFMF.n by A48; then
A74:  (Ir.n) to_power (1/k) <= RF0 to_power (1/k) + (Partial_Sums PAbsFMF).n
        by A47,A73;
     RF0 to_power (1/k) + (Partial_Sums PAbsFMF).n < RF0 to_power (1/k) + Br
        by A72,XREAL_1:8; then
A75: (Ir.n)  to_power (1/k) < RF0 to_power (1/k) + Br by A74,XXREAL_0:2;
     Ir.n = Integral(M,Gp.n) by A38; then
     Ir.n = Integral(M,(G.n) to_power k) by A23; then
A76:Ir.n = Integral(M,abs(G.n) to_power k) by A17;
A77: G.n in Lp_Functions(M,k) by A17; then
     0 <= (Ir.n) to_power (1/k) by Th49,A76,Th4; then
     ((Ir.n) to_power (1/k)) to_power k
       < (RF0 to_power (1/k) + Br) to_power k by A75,Th3; then
     (Ir.n) to_power ((1/k)*k) < (RF0 to_power (1/k) + Br) to_power k
         by A77,Th49,A76,HOLDER_1:2; then
     (Ir.n) to_power 1 < (RF0 to_power (1/k) + Br) to_power k by XCMPLX_1:106;
     hence thesis by POWER:25;
    end; then
A78:Ir is bounded_above by SEQ_2:def 3;
    for n,m be Nat st n <=m holds Ir.n <= Ir.m
    proof
     let n,m be Nat;
     assume n <=m; then
A79:  for x be Element of X st x in E holds (ExtGp.n).x <= (ExtGp.m).x by A28;
A80:  ExtGp.n is E-measurable & ExtGp.m is E-measurable &
     ExtGp.n is nonnegative & ExtGp.m is nonnegative by A26;
A81:  dom(ExtGp.n) = E & dom(ExtGp.m) = E by A26; then
A82: (ExtGp.n)|E = ExtGp.n & (ExtGp.m)|E = ExtGp.m by RELAT_1:68;
     I.n = Integral(M,ExtGp.n) & I.m = Integral(M,ExtGp.m) by A38;
     hence thesis by A79,A81,A80,A82,MESFUNC9:15;
    end; then
    Ir is non-decreasing by SEQM_3:6; then
A83:I is convergent_to_finite_number & lim I = lim Ir by A78,RINFSUP2:14;
    reconsider LExtGp = lim ExtGp as PartFunc of X,ExtREAL;
A84:E = dom LExtGp & LExtGp is E-measurable
       by A26,A27,A37,MESFUNC8:25,def 9;
A85:for x be object st x in dom LExtGp holds 0 <= LExtGp.x
    proof
     let x be object;
     assume A86: x in dom LExtGp; then
     reconsider x1 = x as Element of X;
A87:  x1 in E by A27,A86,MESFUNC8:def 9;
     now let k1 being Nat;
      reconsider k=k1 as Nat;
      ExtGp#x1 is non-decreasing by A35,A87; then
A88:  (ExtGp#x1).0 <= (ExtGp#x1).k by RINFSUP2:7;
      0 <= (ExtGp.0).x1 by A27,SUPINF_2:39;
      hence 0 <= (ExtGp#x1).k1 by A88,MESFUNC5:def 13;
     end; then
     0 <= lim (ExtGp#x1) by A87,A37,MESFUNC9:10;
     hence thesis by A86,MESFUNC8:def 9;
    end;
A89:eq_dom(LExtGp,+infty) = E /\ eq_dom(LExtGp,+infty)
       by A84,RELAT_1:132,XBOOLE_1:28; then
    reconsider EE=eq_dom(LExtGp,+infty) as Element of S by A84,MESFUNC1:33;
    reconsider E0= E \ EE as Element of S;
    E0` = (X \ E) \/ (X /\ EE) by XBOOLE_1:52; then
A90:E0` = E` \/ EE by XBOOLE_1:28;
    M.EE = 0 by A38,A83,A84,A85,A89,MESFUNC9:13,SUPINF_2:52; then
A91:EE is measure_zero of M by MEASURE1:def 7;
    E` is Element of S by MEASURE1:34; then
    E` is measure_zero of M by A4,MEASURE1:def 7; then
    E0` is measure_zero of M by A90,A91,MEASURE1:37; then
A92:M.E0` = 0 by MEASURE1:def 7;
A93:for x be Element of X st x in E0 holds LExtGp.x in REAL
    proof
     let x be Element of X;
     assume x in E0; then
     x in E & not x in EE by XBOOLE_0:def 5; then
     LExtGp.x <> +infty & 0<= LExtGp.x by A84,A85,MESFUNC1:def 15;
     hence LExtGp.x in REAL by XXREAL_0:14;
    end;
A94:for x be Element of X st x in E0 holds
     Gp#x is convergent & lim (Gp#x)= lim (ExtGp#x)
    proof
     let x be Element of X;
     assume A95: x in E0; then
A96:  x in E by XBOOLE_0:def 5; then
     LExtGp.x = lim((ExtGp)#x) by A84,MESFUNC8:def 9; then
A97:  lim((ExtGp)#x) in REAL by A93,A95;
     (ExtGp)#x is convergent by A37,A96; then
A98:  ex g be Real st lim((ExtGp)#x) = g &
      (for p be Real st 0<p ex n be Nat st
        for m be Nat st n<=m holds |. ((ExtGp)#x).m - lim((ExtGp)#x) .| < p) &
      (ExtGp)#x is convergent_to_finite_number
        by A97,MESFUNC5:def 12;
     ExtGp#x = Gp#x by MESFUN7C:1;
     hence thesis by A98,RINFSUP2:15;
    end;
A99:for x be Element of X st x in E0 holds
     for n be Nat holds (Gp#x).n = ((G#x).n) to_power k
    proof
     let x be Element of X;
     assume A100: x in E0;
     hereby let n be Nat;
      x in E by A100,XBOOLE_0:def 5; then
      x in dom(G.n) by A17; then
A101:   x in dom((G.n) to_power k) by MESFUN6C:def 4;
      (Gp#x).n = (Gp.n).x by SEQFUNC:def 10
              .= ((G.n) to_power k ).x by A23
              .= ((G.n).x) to_power k by A101,MESFUN6C:def 4;
      hence (Gp#x).n = ((G#x).n) to_power k by SEQFUNC:def 10;
     end;
    end;
A102:for x be Element of X st x in E0 holds
      (Partial_Sums AbsFMF)#x is convergent
    proof
     let x be Element of X;
     assume A103: x in E0; then
A104:  Gp#x is convergent by A94;
A105:  now let n be Nat;
      G.n is nonnegative by A17; then
      0 <= (G.n).x by MESFUNC6:51; hence
      0 <= (G#x).n by SEQFUNC:def 10;
     end;
     for n be Nat holds
       (Gp#x).n = ((G#x).n) to_power k by A103,A99; then
A106:  G#x is convergent by A104,A105,Th9;
     now let s be Real;
      assume 0<s; then
      consider n be Nat such that
A107:    for m be Nat st n<=m holds |.(G#x).m -(G#x).n qua Complex.|<s
         by A106,SEQ_4:41;
      now let m be Nat;
       assume A108: n<=m;
       x in E by A103,XBOOLE_0:def 5; then
A109:    x in dom(G.n) & x in dom(G.m) by A17;
       G.m = abs(F1.0) + (Partial_Sums AbsFMF).m &
       G.n = abs(F1.0) + (Partial_Sums AbsFMF).n by A16; then
       (G.m).x = abs(F1.0).x + ((Partial_Sums AbsFMF).m).x &
       (G.n).x = abs(F1.0).x + ((Partial_Sums AbsFMF).n).x
          by A109,VALUED_1:def 1; then
       (G#x).m = abs(F1.0).x + ((Partial_Sums AbsFMF).m).x &
       (G#x).n = abs(F1.0).x + ((Partial_Sums AbsFMF).n).x
         by SEQFUNC:def 10; then
A110:    (G#x).m -(G#x).n
        = ((Partial_Sums AbsFMF).m).x -((Partial_Sums AbsFMF).n).x;
       ((Partial_Sums AbsFMF)#x).m = ((Partial_Sums AbsFMF).m).x &
       ((Partial_Sums AbsFMF)#x).n = ((Partial_Sums AbsFMF).n).x
          by SEQFUNC:def 10;
       hence |.((Partial_Sums AbsFMF)#x).m -((Partial_Sums AbsFMF)#x).n
             qua Complex .| < s
         by A107,A108,A110;
      end;
      hence ex n be Nat st for m be Nat st n<=m holds
        |.((Partial_Sums AbsFMF)#x).m - ((Partial_Sums AbsFMF)#x).n
              qua Complex.| < s;
     end;
     hence thesis by SEQ_4:41;
    end;
A111:for x be Element of X st x in E0 holds
     Partial_Sums abs(FMF#x) = (Partial_Sums AbsFMF)#x
    proof
     let x be Element of X;
     assume x in E0; then
A112:  x in E by XBOOLE_0:def 5;
     defpred PQ[Nat] means
      (Partial_Sums abs(FMF#x)).$1 = ((Partial_Sums AbsFMF)#x).$1;
     (Partial_Sums abs(FMF#x)).0 = abs(FMF#x).0 by SERIES_1:def 1
      .= |.(FMF#x).0 .| by VALUED_1:18
      .= |.(FMF.0).x.| by SEQFUNC:def 10
      .= (abs(FMF.0)).x by VALUED_1:18
      .= (AbsFMF.0).x by SEQFUNC:def 4
      .= ((Partial_Sums AbsFMF).0).x by MESFUN9C:def 2
      .= ((Partial_Sums AbsFMF)#x).0 by SEQFUNC:def 10; then
A113:  PQ[ 0];
A114:  now let n be Nat;
      assume A115: PQ[n];
A116:   (Partial_Sums AbsFMF).(n+1) = (Partial_Sums AbsFMF).n + AbsFMF.(n+1)
         by MESFUN9C:def 2;
      dom(AbsFMF.0) = E by A13; then
A117:   x in dom((Partial_Sums AbsFMF).(n+1)) by A112,MESFUN9C:11;
A118:   abs(FMF#x).(n+1) = |.(FMF#x).(n+1).| by VALUED_1:18
       .= |.(FMF.(n+1)).x.| by SEQFUNC:def 10
       .= abs(FMF.(n+1)).x by VALUED_1:18
       .= (AbsFMF.(n+1)).x by SEQFUNC:def 4;
      (Partial_Sums abs(FMF#x)).(n+1)
        = (Partial_Sums abs(FMF#x)).n + abs(FMF#x).(n+1) by SERIES_1:def 1
       .= ((Partial_Sums AbsFMF).n).x + (AbsFMF.(n+1)).x
             by A115,A118,SEQFUNC:def 10
       .= ((Partial_Sums AbsFMF).(n+1)).x by A116,A117,VALUED_1:def 1
       .= ((Partial_Sums AbsFMF)#x).(n+1) by SEQFUNC:def 10;
      hence PQ[n+1];
     end;
     for n be Nat holds PQ[n] from NAT_1:sch 2(A113,A114);
     then for n be Element of NAT holds PQ[n];
     hence thesis by FUNCT_2:63;
    end;
A119:for x be Element of X st x in E0 for n be Nat holds
      (F1#x).(n+1) = (F1#x).0 + (Partial_Sums (FMF#x)).n
    proof
     let x be Element of X;
     assume x in E0; then
A120:x in E by XBOOLE_0:def 5;
     defpred PQ[Nat] means
      (F1#x).($1+1) = (F1#x).0 + (Partial_Sums(FMF#x)).$1;
     dom (FMF.0) = E by A11; then
A121:  x in dom (F1.(0+1) - F1.0) by A10,A120;
     (Partial_Sums (FMF#x)).0 = (FMF#x).0 by SERIES_1:def 1
       .= (FMF.0).x by SEQFUNC:def 10
       .= ( F1.(0+1) - F1.0 ).x by A10; then
A122:  (Partial_Sums (FMF#x)).0 = (F1.(0+1)).x - (F1.0).x by A121,VALUED_1:13;
     (F1#x).0 + (Partial_Sums(FMF#x)).0
       = (F1.0).x + (Partial_Sums (FMF#x)).0 by SEQFUNC:def 10; then
 A123: PQ[ 0] by A122,SEQFUNC:def 10;
 A124: now let n be Nat;
      assume A125:PQ[n];
      dom (FMF.(n+1)) = E by A11; then
A126:  x in dom (F1.((n+1)+1) - F1.(n+1)) by A10,A120;
      (FMF#x).(n+1) = (FMF.(n+1)).x by SEQFUNC:def 10
       .= ((F1.((n+1)+1) - F1.(n+1)).x) by A10; then
A127:   (FMF#x).(n+1) = (F1.((n+1)+1)).x  - (F1.(n+1)).x by A126,VALUED_1:13;
      (F1#x).0 + (Partial_Sums (FMF#x)).(n+1)
       = (F1#x).0 + ((Partial_Sums(FMF#x)).n + (FMF#x).(n+1)) by SERIES_1:def 1
      .= (F1#x).0 + (Partial_Sums (FMF#x)).n + (FMF#x).(n+1)
      .= (F1.(n+1)).x + (FMF#x).(n+1) by A125,SEQFUNC:def 10;
      hence PQ[n+1] by A127,SEQFUNC:def 10;
     end;
     for n be Nat holds PQ[n] from NAT_1:sch 2(A123,A124);
     hence thesis;
    end;
A128:for x be Element of X st x in E0 holds F1#x is convergent
    proof
     let x be Element of X;
     assume A129: x in E0; then
     Partial_Sums abs(FMF#x) = (Partial_Sums AbsFMF)#x by A111; then
     Partial_Sums abs(FMF#x) is convergent by A129,A102; then
     abs(FMF#x) is summable by SERIES_1:def 2; then
     FMF#x is absolutely_summable by SERIES_1:def 4; then
     FMF#x is summable; then
A130:(Partial_Sums (FMF#x)) is convergent by SERIES_1:def 2;
     now let s be Real;
      assume 0<s; then
      consider n be Nat such that
A131:  for m be Nat st n<=m holds
        |.(Partial_Sums (FMF#x)).m -(Partial_Sums (FMF#x)).n qua Complex.|<s
          by A130,SEQ_4:41;
      set n1=n+1;
      now let m1 be Nat;
       assume A132: n1<=m1;
       1 <=n1 by NAT_1:11; then
       reconsider m=m1-1 as Nat by A132,NAT_1:21,XXREAL_0:2;
       n1-1<=m1-1 by A132,XREAL_1:9; then
A133:   |.(Partial_Sums (FMF#x)).m -(Partial_Sums (FMF#x)).n.|<s by A131;
       m1=m+1; then
       (F1#x).(n1) = (F1#x).0 + (Partial_Sums (FMF#x)).n &
       (F1#x).(m1) = (F1#x).0 + (Partial_Sums (FMF#x)).m by A119,A129;
       hence |.(F1#x).m1-(F1#x).n1 qua Complex.|<s by A133;
      end;
      hence ex n be Nat st
             for m be Nat st n<=m holds |.(F1#x).m-(F1#x).n qua Complex.|<s;
     end;
     hence thesis by SEQ_4:41;
    end;
    set F2 = F1||E0;
A134:for x be Element of X st x in E0 holds F2#x is convergent
    proof
     let x be Element of X;
     assume A135: x in E0; then
     F1#x is convergent by A128;
     hence thesis by A135,MESFUN9C:1;
    end;
A136:for x be Element of X st x in E0 holds F2#x = F1#x
    proof
     let x be Element of X;
     assume A137:x in E0;
     now let n be Element of NAT;
      (F2#x).n = (F2.n).x by SEQFUNC:def 10
       .= ((F1.n)|E0).x by MESFUN9C:def 1
       .= (F1.n).x by A137,FUNCT_1:49;
      hence (F2#x).n = (F1#x).n by SEQFUNC:def 10;
     end;
     hence thesis by FUNCT_2:63;
    end;
A138:for n be Nat holds dom(F2.n) = E0 & F2.n is E0-measurable
    proof
     let n be Nat;
A139: dom(F1.0) = E by A7;
     dom (F2.n) = dom((F1.n)|E0) by MESFUN9C:def 1; then
     dom (F2.n) =dom(F1.n) /\ E0 by RELAT_1:61; then
     dom (F2.n) = E /\ E0 by A7;
     hence dom(F2.n) = E0 by XBOOLE_1:28,36;
     for m be Nat holds F1.m is E0-measurable
     proof
      let m be Nat;
      F1.m is E-measurable by A7;
      hence F1.m is E0-measurable by MESFUNC6:16,XBOOLE_1:36;
     end;
     hence F2.n is E0-measurable by A139,MESFUN9C:4,XBOOLE_1:36;
    end;
    reconsider F2 as with_the_same_dom Functional_Sequence of X,REAL
      by MESFUN9C:2;
A140:for n be Nat holds F2.n in Lp_Functions(M,k) & F2.n in Sq.(N.n)
    proof
     let n1 be Nat;
     F2.n1 = (F1.n1)|E0 by MESFUN9C:def 1; then
     abs(F2.n1) = (abs(F1.n1))|E0 by Th13; then
A141: ((abs (F1.n1)) to_power k)|E0 = (abs (F2.n1)) to_power k by Th20;
A142:  F2.n1 is E0-measurable & dom(F2.n1) = E0 by A138;
     F1.n1 in Lp_Functions(M,k) by A7; then
     ex FMF be PartFunc of X,REAL st
      F1.n1= FMF & ex ND be Element of S st M.ND` =0 & dom FMF = ND &
      FMF is ND-measurable & (abs FMF) to_power k is_integrable_on M; then
     ((abs (F2.n1)) to_power k) is_integrable_on M by A141,MESFUNC6:91;
     hence A143: F2.n1 in Lp_Functions(M,k) by A142,A92;
     reconsider n=n1 as Nat;
     set m = N.n;
     F1.n = Fsq.m by A6; then
A144:  F1.n in Sq.(N.n) & Sq.(N.n) = a.e-eq-class_Lp(F1.n,M,k) by A3;
     reconsider EB = E0` as Element of S by MEASURE1:34;
     (F2.n)|EB` = F2.n by A142,RELAT_1:68; then
     (F2.n)|EB` =(F1.n)|EB` by MESFUN9C:def 1; then
     (F2.n) a.e.= (F1.n),M by A92;
     hence thesis by A143,A144,Th36;
    end;
A145:dom (lim F2) = dom (F2.0) by MESFUNC8:def 9; then
A146:dom (lim F2) = E0 by A138;
A147:for x be Element of X st x in E0 holds (lim F2).x = lim (F2#x)
    proof
     let x be Element of X;
     assume x in E0; then
     (lim F2).x =lim R_EAL(F2#x) & F2#x is convergent by A146,A134,MESFUN7C:14;
     hence (lim F2).x = lim (F2#x) by RINFSUP2:14;
    end;
    now let y be object;
     assume y in rng (lim F2); then
     consider x be Element of X such that
A148:   x in dom (lim F2) & y= (lim F2).x by PARTFUN1:3;
     y = lim (F2#x) by A148,A146,A147;
     hence y in REAL by XREAL_0:def 1;
    end; then
    rng (lim F2) c= REAL; then
    reconsider F = lim F2 as PartFunc of X,REAL by A145,RELSET_1:4;
A149:dom(LExtGp|E0) = E /\ E0 by A84,RELAT_1:61; then
A150:dom(LExtGp|E0) = E0 by XBOOLE_1:28,36;
    now let y be object;
     assume y in rng (LExtGp|E0); then
     consider x be Element of X such that
A151:  x in dom (LExtGp|E0) & y=(LExtGp|E0).x by PARTFUN1:3;
     y =(LExtGp).x by A150,A151,FUNCT_1:49;
     hence y in REAL by A150,A151,A93;
    end; then
    rng (LExtGp|E0) c= REAL; then
    reconsider gp = LExtGp|E0 as PartFunc of X,REAL by A149,RELSET_1:4;
A152:for x be Element of X st x in E0 holds gp.x = lim (Gp#x)
    proof
     let x be Element of X;
     assume A153: x in E0; then
     x in dom LExtGp by A84,XBOOLE_0:def 5; then
     LExtGp.x = lim((ExtGp)#x) by MESFUNC8:def 9; then
     gp.x = lim((ExtGp)#x) by A153,FUNCT_1:49;
     hence gp.x = lim (Gp#x) by A94,A153;
    end;
A154:   LExtGp is nonnegative by A85,SUPINF_2:52;
    Integral(M,LExtGp) in REAL by A83,A38,XREAL_0:def 1;
    then LExtGp is_integrable_on M by A154,A84,Th2; then
    R_EAL gp is_integrable_on M by MESFUNC5:97; then
A155:gp is_integrable_on M;
A156:dom(F2.0) = E0 by A138; then
A157:dom F = E0 by MESFUNC8:def 9; then
A158:E0 =dom abs F by VALUED_1:def 11; then
A159:E0 = dom((abs F) to_power k) by MESFUN6C:def 4;
A160:for x be Element of X, n be Nat st x in E0 holds
      |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
          <= (G#x).n
    proof
     let x be Element of X,n be Nat;
     assume A161: x in E0; then
     x in E by XBOOLE_0:def 5; then
A162:  x in dom(G.n) by A17;
     G.n = abs(F1.0) + (Partial_Sums AbsFMF).n by A16; then
     (G.n).x = abs(F1.0).x + ((Partial_Sums AbsFMF).n).x
        by A162,VALUED_1:def 1; then
A163:  (G.n).x = |.(F1.0).x qua Complex.| + ((Partial_Sums AbsFMF).n).x
       by VALUED_1:18;
     (G#x).n = (G.n).x by SEQFUNC:def 10
       .=|.(F1.0).x qua Complex.| + ((Partial_Sums AbsFMF)#x).n
              by A163,SEQFUNC:def 10
       .=|.(F1#x).0 qua Complex .| + ((Partial_Sums AbsFMF)#x).n
         by SEQFUNC:def 10; then
A164:  (G#x).n =|.(F1#x).0 qua Complex .| + (Partial_Sums abs(FMF#x)).n
           by A111,A161;
     |.(Partial_Sums (FMF#x)).n qua Complex.| <= Partial_Sums abs(FMF#x).n
             by Lm1;
     hence thesis by A164,XREAL_1:6;
    end;
A165:for x be Element of X, n be Nat st x in E0 holds
     |.(F1#x).0 + (Partial_Sums (FMF#x)).n qua Complex.| to_power k <= (Gp#x).n
    proof
     let x be Element of X ,n be Nat;
     assume A166: x in E0; then
A167:  (Gp#x).n = (((G#x)).n) to_power k by A99;
A168: |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
        <= (G#x).n
           by A160,A166;
     |.(F1#x).0 + (Partial_Sums (FMF#x)).n qua Complex.|
       <= |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
            by COMPLEX1:56;
      then
A169: |.(F1#x).0 + (Partial_Sums (FMF#x)).n.| <= (G#x).n by A168,XXREAL_0:2;
     0 <= |.(F1#x).0 + (Partial_Sums (FMF#x)).n.| by COMPLEX1:46;
     hence thesis by A167,A169,HOLDER_1:3;
    end;
A170:for x be Element of X, n be Nat st x in E0 holds
      |.(F2#x).n qua Complex.| to_power k <= (Gp#x).n
    proof
     let x be Element of X, n be Nat;
     assume A171: x in E0; then
A172:  F1#x = F2#x by A136;
     per cases;
     suppose A173:n = 0;
A174:   (Gp#x).n = (((G#x)).n) to_power k by A171,A99;
A175:   |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
            <= (G#x).n by A160,A171;
      0 <= |.(Partial_Sums(FMF#x)).n.| by COMPLEX1:46; then
      0 + |.(F1#x).0 qua Complex .|
     <= |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
           by XREAL_1:6; then
A176:   |.(F1#x).0 qua Complex .| <= (G#x).n by A175,XXREAL_0:2;
      0 <= |.(F1#x).0 .| by COMPLEX1:46;
      hence |.((F2#x).n) qua Complex.| to_power k
     <= (Gp#x).n by A172,A173,A174,A176,
HOLDER_1:3;
     end;
     suppose n <> 0; then
      consider m be Nat such that
A177:    n=m+1 by NAT_1:6;
      reconsider m as Nat;
      (F1#x).(m+1) = (F1#x).0 + (Partial_Sums(FMF#x)).m by A119,A171; then
A178:   |.(F1#x).(m+1) qua Complex.| to_power k <= (Gp#x).m by A165,A171;
      x in E by A171,XBOOLE_0:def 5; then
A179:   ExtGp#x is non-decreasing by A35;
A180:   ((ExtGp)#x).m <= ((ExtGp)#x).(m+1) by A179;
      ExtGp#x = Gp#x by MESFUN7C:1;
      hence |.((F2#x).n) qua Complex.| to_power k <= (Gp#x).n
            by A172,A177,A178,A180,
XXREAL_0:2;
     end;
    end;
A181:for x be Element of X st x in E0
   holds |.((abs F) to_power k).x qua Complex .| <= gp.x
    proof
     let x be Element of X;
     assume A182: x in E0; then
A183:  Gp#x is convergent by A94;
     deffunc ABSF2(set) = (abs(F2#x).$1 ) to_power k;
     consider s be Real_Sequence such that
A184:   for n be Nat holds s.n=ABSF2(n) from SEQ_1:sch 1;
A185:  gp.x = lim (Gp#x) by A152,A182;
A186:  ((abs F) to_power k).x = ((abs F).x) to_power k by A159,A182,
MESFUN6C:def 4
     .= (|.F.x qua Complex.|) to_power k by A158,A182,VALUED_1:def 11
     .= |.lim (F2#x) qua Complex.| to_power k by A182,A147
     .= (lim (abs (F2#x))) to_power k by A134,A182,SEQ_4:14;
A187: now let n be Nat;
      0 <= |.(F2#x).n.| by COMPLEX1:46;
      hence 0 <=(abs(F2#x)).n by VALUED_1:18;
     end;
     abs (F2#x) is convergent by A182,A134,SEQ_4:13; then
A188:  s is convergent & (lim (abs (F2#x))) to_power k = lim s
        by A187,A184,HOLDER_1:10;
     now let n be Nat;
      |.(F2#x).n qua Complex.| to_power k <= (Gp#x).n by A170,A182; then
      (abs(F2#x).n) to_power k <= (Gp#x).n by VALUED_1:18;
      hence s.n <=(Gp#x).n by A184;
     end; then
A189:  ((abs F) to_power k).x <= gp.x by A188,A185,A186,A183,SEQ_2:18;
     0 <= ((abs F) to_power k).x by MESFUNC6:51;
     hence |.((abs F) to_power k).x qua Complex.| <= gp.x
         by A189,ABSVALUE:def 1;
    end;
    R_EAL F is E0-measurable by A138,A156,A134,MESFUN7C:21; then
A190:F is E0-measurable; then
A191:(abs F) is E0-measurable by A157,MESFUNC6:48;
    dom abs F = E0 by A157,VALUED_1:def 11; then
    (abs F) to_power k is E0-measurable by A191,MESFUN6C:29; then
    (abs F) to_power k is_integrable_on M by A150,A155,A159,A181,MESFUNC6:96;
 then
A192:F in Lp_Functions(M,k) by A92,A157,A190;
A193:for x be Element of X, n,m be Nat st x in E0 & m <= n holds
      |. (F1#x).n -(F1#x).m  qua Complex.| to_power k <= (Gp#x).n
    proof
     let x be Element of X, n1,m1 be Nat;
     assume A194: x in E0 & m1 <= n1;
     now per cases;
      suppose A195: m1= 0;
       now per cases;
        suppose A196: n1=0;
         ((G.n1) to_power k) is nonnegative by A24; then
         Gp.n1 is nonnegative by A23; then
         0 <= (Gp.n1).x by MESFUNC6:51; then
         0 <= (Gp#x).n1 by SEQFUNC:def 10;
         hence |. (F1#x).n1-(F1#x).m1 qua Complex .| to_power k <= (Gp#x).n1
            by A195,A196,COMPLEX1:44,POWER:def 2;
        end;
        suppose n1<> 0; then
         consider n be Nat such that
A197:     n1=n+1 by NAT_1:6;
         reconsider n as Nat;
A198:    (F1#x).(n+1) = (F1#x).0 + (Partial_Sums (FMF#x)).n by A194,A119;
A199:      |.(F1#x).0 qua Complex .| + |.(Partial_Sums (FMF#x)).n qua Complex.|
          <= (G#x).n by A160
,A194;
         0 <= |.(F1#x).0 .| by COMPLEX1:46; then
         |.(Partial_Sums (FMF#x)).n qua Complex.| + 0
           <= |.(F1#x).0 qua Complex .| +
           |.(Partial_Sums (FMF#x)).n qua Complex.|
               by XREAL_1:6;
          then
A200:   |.(Partial_Sums (FMF#x)).n qua Complex.| <= (G#x).n by A199,XXREAL_0:2;
         0 <= |.(Partial_Sums (FMF#x)).n.| by COMPLEX1:46; then
A201:    |. (Partial_Sums(FMF#x)).n qua Complex .| to_power k
           <= ((G#x).n) to_power k by A200,HOLDER_1:3;
A202:    (Gp#x).n = (((G#x)).n) to_power k by A194,A99;
         x in E by A194,XBOOLE_0:def 5; then
A203:    ExtGp#x is non-decreasing by A35;
A204:    ((ExtGp)#x).n <= ((ExtGp)#x).(n+1) by A203;
         ExtGp#x = Gp#x by MESFUN7C:1;
         hence |. (F1#x).n1  -(F1#x).m1 qua Complex .| to_power k <= (Gp#x).n1
           by A195,A197,A204,A201,A202,A198,XXREAL_0:2;
        end;
       end;
       hence |. (F1#x).n1-(F1#x).m1 qua Complex .| to_power k <= (Gp#x).n1;
      end;
      suppose A205: m1<> 0; then
       consider m be Nat such that
A206:    m1=m+1 by NAT_1:6;
       reconsider m as Nat;
       0 < n1 by A194,A205; then
       consider n be Nat such that
A207:    n1=n+1 by NAT_1:6;
       reconsider n as Element of NAT by ORDINAL1:def 12;
A208:   m1-1 <= n1-1 by A194,XREAL_1:9;
       x in E by A194,XBOOLE_0:def 5; then
A209:   x in dom(G.n) by A17; then
A210:  x in dom ((G.n) to_power k) by MESFUN6C:def 4;
       (Gp#x).n = (Gp.n).x by SEQFUNC:def 10; then
       (Gp#x).n = ((G.n) to_power k).x by A23; then
A211:  (Gp#x).n = ((G.n).x) to_power k by A210,MESFUN6C:def 4;
       G.n = abs(F1.0) + (Partial_Sums AbsFMF).n by A16; then
       (G.n).x = abs(F1.0).x + ((Partial_Sums AbsFMF).n).x
           by A209,VALUED_1:def 1
       .= |.(F1.0).x qua Complex.| + ((Partial_Sums AbsFMF).n).x
           by VALUED_1:18; then
A212
:   (G.n).x = |.(F1.0).x qua Complex.| + ((Partial_Sums AbsFMF)#x).n
            by SEQFUNC:def 10;
A213:    (F1#x).(n+1) = (F1#x).0 + (Partial_Sums (FMF#x)).n &
       (F1#x).(m+1) = (F1#x).0 + (Partial_Sums (FMF#x)).m by A194,A119;
A214:    |. (Partial_Sums (FMF#x)).n - (Partial_Sums (FMF#x)).m .|
         <= (Partial_Sums abs(FMF#x)).n by Th10,A206,A207,A208;
A215:    (Partial_Sums abs(FMF#x)).n = ((Partial_Sums AbsFMF)#x).n by A111,A194
;
       0 <= |.(F1.0).x.| by COMPLEX1:46; then
       0 + (Partial_Sums abs(FMF#x)).n
         <= |.(F1.0).x qua Complex.| + ((Partial_Sums AbsFMF)#x).n
           by A215,XREAL_1:6; then
A216:   |. (F1#x).(n+1)-(F1#x).(m+1) .| <= (G.n).x by A212,A213,A214,XXREAL_0:2
;
       0 <= |. (F1#x).(n+1)-(F1#x).(m+1) .| by COMPLEX1:46; then
A217:   (|. (F1#x).(n+1)-(F1#x).(m+1) qua Complex .|) to_power k <= (Gp#x).n
           by A211,A216,HOLDER_1:3;
       x in E by A194,XBOOLE_0:def 5; then
A218:  ExtGp#x is non-decreasing by A35;
A219:  ((ExtGp)#x).n <= ((ExtGp)#x).(n+1) by A218;
       ExtGp#x = Gp#x by MESFUN7C:1;
       hence (|. (F1#x).(n1)-(F1#x).(m1) qua Complex .|) to_power k
            <= (Gp#x).n1
          by A206,A207,A219,A217,XXREAL_0:2;
      end;
     end;
     hence thesis;
    end;
A220:for x be Element of X, n be Nat st x in E0 holds
      |. F.x -(F2#x).n qua Complex .| to_power k <= gp.x
    proof
     let x be Element of X, n1 be Nat;
     assume A221: x in E0; then
A222:  Gp#x is convergent by A94;
A223:  F1#x = F2#x by A136,A221;
A224:  F2#x is convergent by A221,A134;
     reconsider n=n1 as Nat;
     deffunc AF2F20(Nat) = (F2#x).$1 -(F2#x).n;
     consider s0 be Real_Sequence  such that
A225:  for j be Nat holds s0.j=AF2F20(j) from SEQ_1:sch 1;
A226: now let p be Real;
      assume 0<p; then
      consider n1 be Nat such that
A227:    for m be Nat st n1<=m holds
         |.(F2#x).m-lim (F2#x) qua Complex.|<p by A224,SEQ_2:def 7;
      take n1;
      thus for m be Nat st n1<=m holds
             |.s0.m-(lim (F2#x)- (F2#x).n) qua Complex.| < p
      proof
       let m be Nat;
       assume A228: n1<=m;
       s0.m-(lim (F2#x)- (F2#x).n)
        = (F2#x).m -(F2#x).n-(lim (F2#x)- (F2#x).n) by A225; then
       s0.m-(lim (F2#x)- (F2#x).n) = (F2#x).m - lim (F2#x);
       hence |.s0.m-(lim (F2#x)- (F2#x).n) qua Complex.|<p by A228,A227;
      end;
     end; then
A229: s0 is convergent by SEQ_2:def 6; then
     lim s0 = lim (F2#x) - (F2#x).n by A226,SEQ_2:def 7; then
A230: lim abs s0 = |.lim (F2#x) - (F2#x).n.| by A229,SEQ_4:14;
A231: abs s0 is convergent  by A229;
     deffunc AF2F2(Nat)= |. (F2#x).$1-(F2#x).n qua Complex .| to_power k;
     consider s be Real_Sequence such that
A232:   for j be Nat holds s.j=AF2F2(j) from SEQ_1:sch 1;
A233: for j be Nat st n <= j holds s.j <= (Gp#x).j
     proof
      let j be Nat;
      assume n <= j; then
      |. (F2#x).j  -(F2#x).n qua Complex .| to_power k <= (Gp#x).j
                 by A223,A221,A193;
      hence thesis by A232;
     end;
A234: now let n be Nat;
      0 <= |.s0.n.| by COMPLEX1:46;
      hence  0 <=(abs(s0)).n by VALUED_1:18;
     end;
     now let j be Nat;
      thus s.j = |. (F2#x).j-(F2#x).n qua Complex .| to_power k by A232
             .= |.s0.j qua Complex.| to_power k by A225
             .= (abs(s0).j) to_power k by VALUED_1:18;
     end; then
A235: s is convergent & lim s = (lim (abs s0)) to_power k
        by A234,A231,HOLDER_1:10; then
A236: s^\n is convergent & lim (s^\n)=lim s by SEQ_4:20;
     gp.x = lim (Gp#x) by A152,A221; then
A237: (Gp#x)^\n is convergent & lim ((Gp#x)^\n) = gp.x by A222,SEQ_4:20;
     for j be Nat holds (s^\n).j <= ((Gp#x)^\n).j
     proof
      let j be Nat;
      (s^\n).j = s.(n+j) & ((Gp#x)^\n).j = (Gp#x).(n+j) by NAT_1:def 3;
      hence thesis by A233,NAT_1:11;
     end; then
     lim s <= gp.x by A236,A237,SEQ_2:18;
     hence thesis by A230,A235,A147,A221;
    end;
    deffunc FX3(Nat) = (|. F -F2.$1 .|) to_power k;
    consider FP be Functional_Sequence of X,REAL such that
A238: for n be Nat holds FP.n = FX3(n) from SEQFUNC:sch 1;
A239:for n be Nat holds dom (FP.n) = E0
    proof
     let n1 be Nat;
     reconsider n=n1 as Nat;
A240:  dom(F2.n) = E0 by A138;
     dom(FP.n1) = dom(abs( F - F2.n ) to_power k) by A238; then
     dom(FP.n1) = dom(abs(F - F2.n)) by MESFUN6C:def 4; then
     dom(FP.n1) = dom (F - F2.n) by VALUED_1:def 11; then
     dom(FP.n1) = E0 /\ E0 by A240,A146,VALUED_1:12;
     hence dom(FP.n1) = E0;
    end; then
A241:E0 = dom(FP.0); then
A242:dom (lim FP) = E0 by MESFUNC8:def 9;
    for n,m be Nat holds dom(FP.n) = dom(FP.m)
    proof
     let n,m be Nat;
     thus dom (FP.n) =E0 by A239 .=dom (FP.m) by A239;
    end; then
    reconsider FP as with_the_same_dom Functional_Sequence of X,REAL
       by MESFUNC8:def 2;
A243:for n be Nat holds FP.n is E0-measurable
    proof
     let n1 be Nat;
     reconsider n=n1 as Nat;
     dom (F2.n) = E0 by A138; then
A244:  dom (F - F2.n) = E0 /\ E0 by A146,VALUED_1:12;
     F2.n is E0-measurable & dom (F2.n) = E0 by A138; then
     F - F2.n is E0-measurable by A190,MESFUNC6:29; then
A245:abs (F - F2.n) is E0-measurable by A244,MESFUNC6:48;
     dom (abs (F - F2.n) ) =E0 by A244,VALUED_1:def 11; then
     (abs (F - F2.n) to_power k) is E0-measurable by A245,MESFUN6C:29;
     hence thesis by A238;
    end;
    for x be Element of X, n be Nat st x in E0 holds (|. FP.n .|).x <= gp.x
    proof
     let x be Element of X, n1 be Nat;
     reconsider n=n1 as Element of NAT by ORDINAL1:def 12;
     assume A246: x in E0; then
A247:  x in dom (FP.n) by A239; then
     x in dom (( |.F -F2.n .| ) to_power k) by A238; then
     x in dom (( |.F -F2.n .| ) ) by MESFUN6C:def 4; then
A248:  x in dom (F -F2.n) by VALUED_1:def 11;
A249: FP.n1 = (|. F -F2.n1 .|) to_power k by A238;
A250:0 <= (|.F.x -(F2.n1).x qua Complex .|) to_power k by Th4,COMPLEX1:46;
     (|. FP.n .|).x =(|. (FP.n).x .|) by VALUED_1:18
     .= |. (|. F -F2.n1 .|.x) to_power k .| by A247,A249,MESFUN6C:def 4
     .= |. (|. (F -F2.n1).x qua Complex .|) to_power k .| by VALUED_1:18
     .= |. (|. F.x -(F2.n1).x qua Complex .|) to_power k .| by A248,VALUED_1:13
     .= (|. F.x -(F2.n1).x qua Complex .|) to_power k by A250,ABSVALUE:def 1
     .= |. F.x - (F2#x).n qua Complex .| to_power k by SEQFUNC:def 10;
     hence thesis by A220,A246;
    end; then
    consider Ip be Real_Sequence such that
A251: (for n be Nat holds Ip.n = Integral(M,FP.n)) &
     ( (for x be Element of X st x in E0 holds FP#x is convergent)
         implies Ip is convergent & lim Ip = Integral(M,lim FP) )
            by A150,A155,A241,A243,MESFUN9C:48;
A252:for x be Element of X st x in E0 holds FP#x is convergent & lim (FP#x) = 0
    proof
     let x be Element of X;
     assume A253: x in E0;
A254:  for n be Nat holds
      (FP#x).n = ( |.lim (F2#x) - (F2#x).n qua Complex .|) to_power k
     proof
      let n be Nat;
      x in dom (FP.n) by A253,A239; then
A255:   x in dom (( |.F -F2.n .| ) to_power k) by A238; then
      x in dom ( |.F -F2.n .| ) by MESFUN6C:def 4; then
A256:   x in dom (F -F2.n) by VALUED_1:def 11;
      thus
      (FP#x).n =(FP.n).x by SEQFUNC:def 10
      .= (( |.F -F2.n .| ) to_power k).x by A238
      .= ( (|.F -F2.n .|).x ) to_power k by A255,MESFUN6C:def 4
      .= ( |.(F -F2.n).x qua Complex .|) to_power k by VALUED_1:18
      .= ( |.(F.x -(F2.n).x) qua Complex .|) to_power k by A256,VALUED_1:13
      .= ( |.(lim (F2#x) -(F2.n).x) qua Complex .|) to_power k by A147,A253
      .= ( |.(lim (F2#x) -(F2#x).n) qua Complex .|) to_power k
               by SEQFUNC:def 10;
     end;
     F2#x is convergent by A253,A134;
     hence thesis by A254,Th11;
    end;
A257:for x be Element of X st x in dom (lim FP) holds 0= (lim FP).x
    proof
     let x be Element of X;
     assume A258: x in dom (lim FP); then
A259: lim (FP#x) = 0 & (FP#x) is convergent by A252,A242;
     (lim FP).x =lim R_EAL(FP#x) by A258,MESFUN7C:14;
     hence thesis by A259,RINFSUP2:14;
    end;
    a.e-eq-class_Lp(F,M,k) in CosetSet(M,k) by A192; then
    reconsider Sq0= a.e-eq-class_Lp(F,M,k) as Point of Lp-Space(M,k)
       by Def11;
A260:for n be Nat holds Ip.n = (||.Sq0-Sq.(N.n).||) to_power k
    proof
     let n be Nat;
     set m = N.n;
     reconsider n1=n as Nat;
A261:  FP.n = abs( F - F2.n1 ) to_power k by A238;
A262:  F in Lp_Functions(M,k) & F in Sq0 by A192,Th38;
     F2.n1 in Lp_Functions(M,k) & F2.n1 in (Sq.m) by A140; then
     (-1)(#)(F2.n1) in (-1)*(Sq.m) by Th54; then
     F -(F2.n1) in (Sq0) + (-1)*(Sq.m) by Th54,A262; then
     F -(F2.n1) in (Sq0) - (Sq.m) by RLVECT_1:16; then
     consider r be Real such that
A263:   0<= r & r = Integral(M,(abs (F -(F2.n1)) ) to_power k) &
      ||. Sq0 - (Sq.m) .|| =r to_power (1/k) by Th53;
     ||. Sq0 - (Sq.m) .|| to_power k = r to_power ((1/k)*k)  by A263,HOLDER_1:2
     .= r to_power 1 by XCMPLX_1:106
     .= r by POWER:25;
     hence thesis by A263,A261,A251;
    end;
    deffunc UZ(Nat) = ||.Sq0-Sq.(N.$1).||;
    consider Iq be Real_Sequence such that
A264:for n be Nat holds Iq.n = UZ(n) from SEQ_1:sch 1;
A265: for n being Nat holds Iq.n = ||.Sq0-Sq.(N.n).|| by A264;
    Iq is convergent & lim Iq = 0
    proof
A266: for n holds Ip.n >= 0
     proof
      let n;
      ||. Sq0-Sq.(N.n) .|| to_power k >= 0 by Th4;
      hence Ip.n >= 0 by A260;
     end;
A267:  for n be Nat holds Iq.n = (Ip.n) to_power (1/k)
     proof
      let n be Nat;
      thus (Ip.n) to_power (1/k)
       = ( ||.Sq0-Sq.(N.n).|| to_power k) to_power (1/k) by A260
      .= ||.Sq0-Sq.(N.n).|| to_power (k*(1/k)) by HOLDER_1:2
      .= (||.Sq0-Sq.(N.n).||) to_power 1 by XCMPLX_1:106
      .= ||.Sq0-Sq.(N.n).|| by POWER:25
      .= Iq.n by A265;
     end;
     hence Iq is convergent by A266,A252,A251,HOLDER_1:10;
     lim Iq = (lim Ip) to_power (1/k) by A252,A251,A266,A267,HOLDER_1:10; then
     lim Iq = 0 to_power (1/k) by A252,A251,A257,A242,LPSPACE1:22;
     hence lim Iq = 0 by POWER:def 2;
    end;
    hence thesis by A2,A265,Lm7;
end;
