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 Th63:
for Sq be sequence of Lp-Space(M,k) holds
 ex Fsq be with_the_same_dom Functional_Sequence of X,REAL st
  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)
proof
    let Sq be sequence of Lp-Space(M,k);
    consider Fsq be Functional_Sequence of X,REAL such that
A1:  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 r = Integral(M,(abs (Fsq.n)) to_power k) &
                      ||. Sq.n .|| = r to_power (1/k) by Th62;
    defpred P[Nat,set] means
     ex DMFSQN be Element of S st $2 = DMFSQN &
      ex FSQN be PartFunc of X,REAL st
       Fsq.$1 = FSQN & M.DMFSQN` = 0 & dom FSQN = DMFSQN &
       FSQN is DMFSQN-measurable &
       (abs FSQN) to_power k is_integrable_on M;
A2: for n being Element of NAT ex y being Element of S st P[n,y]
    proof
     let n be Element of NAT;
     Fsq.n in Lp_Functions(M,k) by A1; then
     ex FMF be PartFunc of X,REAL st Fsq.n = 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;
     hence thesis;
    end;
    consider G be sequence of S such that
A3:  for n be Element of NAT holds P[n,G.n] from FUNCT_2:sch 3(A2);
    reconsider E0 = meet rng G as Element of S;
A4: for n be Nat holds M.(X \ G.n) =0 & E0 c= dom(Fsq.n)
    proof
     let n be Nat;
A5:   n in NAT by ORDINAL1:def 12;
     ex D be Element of S st G.n=D & ex F be PartFunc of X,REAL st
       Fsq.n= F & M.D` =0 & dom F = D & F is D-measurable &
       (abs F) to_power k is_integrable_on M by A3,A5;
     hence M.(X \ (G.n)) =0 & E0 c= dom(Fsq.n) by FUNCT_2:4,SETFAM_1:3,A5;
    end;
A6: (X \ rng G) is N_Sub_set_fam of X by MEASURE1:21;
    for A be set st A in (X \ rng G) holds A in S & A is measure_zero of M
    proof
     let A being set;
     assume A7: A in (X \ rng G); then
     reconsider A0 = A as Subset of X;
     A0` in rng G by A7,SETFAM_1:def 7; then
     consider n be object such that
A8:  n in NAT & A0` = G.n by FUNCT_2:11;
     reconsider n as Nat by A8;
A9: (A0`)` = A0; then
     A0 = X \ G.n by A8;
     hence A in S by MEASURE1:34;
A10: M.A0= 0 by A4,A8,A9;
     A0= X \ (G.n) by A8,A9; then
     A is Element of S by MEASURE1:34;
     hence A is measure_zero of M by A10,MEASURE1:def 7;
    end; then
A11: (for A be object st A in (X \ rng G) holds A in S) &
    (for A be set st A in (X \ rng G) holds A is measure_zero of M); then
    (X \ rng G) c= S; then
    (X \ rng G) is N_Measure_fam of S by A6,MEASURE2:def 1; then
A12: union (X \ rng G) is measure_zero of M by A11,MEASURE2:14;
    E0` = X \ (X \ (union (X \ rng G))) by MEASURE1:4
       .= X /\ union (X \ rng G) by XBOOLE_1:48
       .= union (X \ rng G) by XBOOLE_1:28; then
A13: M.E0` =0 by A12,MEASURE1:def 7;
    set Fsq2 = Fsq||E0;
A14: for n be Nat holds dom (Fsq2.n) = E0
    proof
     let n be Nat;
     dom (Fsq2.n) = dom((Fsq.n)|E0) by MESFUN9C:def 1; then
     dom (Fsq2.n) =dom(Fsq.n) /\ E0 by RELAT_1:61;
     hence dom (Fsq2.n) = E0 by A4,XBOOLE_1:28;
    end;
    now let n,m be Nat;
     dom(Fsq2.n) =E0 & dom(Fsq2.m) = E0 by A14;
     hence dom(Fsq2.n) = dom(Fsq2.m);
    end; then
    reconsider Fsq2 as with_the_same_dom Functional_Sequence of X,REAL
      by MESFUNC8:def 2;
    take Fsq2;
    hereby let n be Nat;
     Fsq.n in Lp_Functions(M,k) by A1; then
A15:  ex FMF be PartFunc of X,REAL st Fsq.n= 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 reconsider E2 = dom (Fsq.n) as Element of S;
A16:  E2 /\ E0 = E0 by A4,XBOOLE_1:28;
     R_EAL (Fsq.n) is E2-measurable by A15; then
     R_EAL (Fsq.n) is E0-measurable by A4,MESFUNC1:30; then
     Fsq.n is E0-measurable; then
     (Fsq.n)|E0 is E0-measurable by A16,MESFUNC6:76; then
A17:  Fsq2.n is E0-measurable by MESFUN9C:def 1;
A18: dom (Fsq2.n)= E0 by A14;
     dom ((abs (Fsq.n)) to_power k) = dom abs (Fsq.n) &
     dom ((abs (Fsq2.n)) to_power k) = dom abs (Fsq2.n) by MESFUN6C:def 4; then
A19: dom ((abs (Fsq.n)) to_power k) = dom (Fsq.n) &
     dom ((abs (Fsq2.n)) to_power k) = dom (Fsq2.n) by VALUED_1:def 11;
     for x be object st x in dom((abs (Fsq2.n)) to_power k) holds
       ((abs (Fsq2.n)) to_power k ).x = ((abs (Fsq.n)) to_power k).x
     proof
      let x be object;
      assume A20: x in dom((abs (Fsq2.n)) to_power k); then
      reconsider x0=x as Element of X;
A21:   x in dom((abs (Fsq.n)) to_power k) by A18,A19,A16,A20,XBOOLE_0:def 4;
      thus ((abs (Fsq2.n)) to_power k ).x
        = ((abs (Fsq2.n)).x0) to_power k by A20,MESFUN6C:def 4
       .= |.((Fsq2.n).x0) qua Complex.| to_power k by VALUED_1:18
       .= |.(((Fsq.n)|E0).x0) qua Complex.| to_power k by MESFUN9C:def 1
       .= |.((Fsq.n).x0) qua Complex.| to_power k by A18,A19,A20,FUNCT_1:49
       .= (abs(Fsq.n).x0) to_power k by VALUED_1:18
       .= ((abs (Fsq.n)) to_power k).x by A21,MESFUN6C:def 4;
     end; then
     ((abs (Fsq.n)) to_power k)|E0 = (abs (Fsq2.n)) to_power k
        by A14,A16,A19,FUNCT_1:46; then
     ((abs (Fsq2.n)) to_power k) is_integrable_on M by A15,MESFUNC6:91;
     hence
A22: Fsq2.n in Lp_Functions(M,k) by A17,A18,A13;
A23: Fsq.n in Sq.n & Sq.n= a.e-eq-class_Lp(Fsq.n,M,k) by A1;
     reconsider EB = E0` as Element of S by MEASURE1:34;
     (Fsq2.n)|EB` = Fsq2.n by A18,RELAT_1:68; then
     (Fsq2.n)|EB` =(Fsq.n)|EB` by MESFUN9C:def 1; then
A24: (Fsq2.n) a.e.= (Fsq.n),M by A13;
     hence Fsq2.n in Sq.n by A23,A22,Th36;
     a.e-eq-class_Lp((Fsq2.n),M,k) = a.e-eq-class_Lp((Fsq.n),M,k)
        by Th42,A24;
     hence Sq.n= a.e-eq-class_Lp(Fsq2.n,M,k) by A1;
     hence ex r be Real st
       0 <= r & r = Integral(M, (abs (Fsq2.n)) to_power k) &
       ||. Sq.n .|| =r to_power (1/k) by Th53,Th38,A22;
    end;
end;
