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 Th62:
for Sq be sequence of Lp-Space(M,k) holds
 ex Fsq be 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
     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);
   defpred P[Nat,set ] means
    ex f be PartFunc of X,REAL st $2=f & f in Lp_Functions(M,k) &
      f in Sq.$1 & Sq.$1= a.e-eq-class_Lp(f,M,k) &
      ex r be Real st
        r = Integral(M,(abs f) to_power k) & ||. Sq.$1 .|| =r to_power (1/k);
A1:for x being Element of NAT ex y being Element of PFuncs(X,REAL) st P[x,y]
   proof
    let x be Element of NAT;
    consider y be PartFunc of X,REAL such that
A2: y in Lp_Functions(M,k) & Sq.x= a.e-eq-class_Lp(y,M,k) by Th53;
    ex r be Real st 0 <= r &
      r = Integral(M,(abs y) to_power k) & ||. Sq.x .|| =r to_power (1/k)
        by Th53,A2,Th38;
    hence thesis by A2,Th38;
   end;
   consider G be sequence of PFuncs(X,REAL) such that
A3: for n be Element of NAT holds P[n,G.n] from FUNCT_2:sch 3(A1);
   reconsider G as Functional_Sequence of X,REAL;
   now let n be Nat;
    n in NAT by ORDINAL1:def 12;
    then
    ex f be PartFunc of X,REAL st G.n=f & f in Lp_Functions(M,k) & f in Sq.n &
     Sq.n= a.e-eq-class_Lp(f,M,k) &
     ex r be Real st
      r = Integral(M,(abs f) to_power k) &
      ||. Sq.n .|| =r to_power (1/k) by A3;
    hence
     G.n in Lp_Functions(M,k) & G.n in Sq.n & Sq.n= a.e-eq-class_Lp(G.n,M,k) &
     ex r be Real st
      r = Integral(M,(abs (G.n)) to_power k) & ||. Sq.n .|| =r to_power (1/k);
   end;
   hence thesis;
end;
