
theorem Lm814C:
  for X be RealNormSpace, f be sequence of DualSp X, x be sequence of X
    st ||.f.|| is bounded holds
  ex F be sequence of Funcs(NAT,the carrier of DualSp X),
     N be sequence of Funcs(NAT,NAT) st
     F.0 is subsequence of f & (F.0)#(x.0) is convergent
   & N.0 is increasing sequence of NAT & F.0 = f*N.0
   & (for k be Nat holds F.(k+1) is subsequence of F.k)
   & (for k be Nat holds (F.(k+1))#(x.(k+1)) is convergent)
   & (for k be Nat holds (F.(k+1))#(x.(k+1)) is subsequence of (F.k)#(x.(k+1)))
   & (for k be Nat holds N.(k+1) is increasing sequence of NAT)
   & for k be Nat holds F.(k+1) = (F.k)*N.(k+1)
proof
  let X be RealNormSpace, f be sequence of DualSp X, x be sequence of X;
  assume ||.f.|| is bounded; then
  consider f0 be sequence of DualSp X such that
P0: f0 is subsequence of f & ||.f0.|| is bounded
  & f0#(x.0) is convergent & f0#(x.0) is subsequence of f#(x.0) by Lm814A1;
  consider N0 be increasing sequence of NAT such that
R0: f0 = f*N0 by VALUED_0:def 17,P0;
  set D1 = Funcs(NAT,the carrier of DualSp X);
  set D2 = Funcs(NAT,NAT);
  reconsider A=f0 as Element of D1 by FUNCT_2:8;
  reconsider B=N0 as Element of D2 by FUNCT_2:8;
  defpred P[Nat, sequence of DualSp X, sequence of NAT,
            sequence of DualSp X, sequence of NAT] means
    ||.$2.|| is bounded implies
       $4 is subsequence of $2 & ||.$4.|| is bounded
     & ($4)#(x.($1+1)) is convergent
     & ($4)#(x.($1+1)) is subsequence of ($2)#(x.($1+1))
     & $5 is increasing sequence of NAT & $4 = $2*$5;
P1: for n being Nat for z being Element of D1, y being Element of D2
      ex z1 being Element of D1, y1 being Element of D2 st P[n,z,y,z1,y1]
  proof
    let n be Nat;
    let z be Element of D1, y be Element of D2;
    consider f0 be sequence of DualSp X, N be increasing sequence of NAT
      such that
X1:   ||.z.|| is bounded implies
        f0 is subsequence of z & ||.f0.|| is bounded
      & f0#(x.(n+1)) is convergent
      & f0#(x.(n+1)) is subsequence of z#(x.(n+1))
      & f0 = z*N by Lm814A2;
    reconsider z1=f0 as Element of D1 by FUNCT_2:8;
    reconsider y1=N as Element of D2 by FUNCT_2:8;
    take z1,y1;
    thus thesis by X1;
  end;
  consider F be sequence of D1, N be sequence of D2 such that
X2: F.0 = A & N.0 = B &
    for n being Nat holds P[n, F.n, N.n, F.(n+1), N.(n+1)]
      from RECDEF_2:sch 3(P1);
  defpred Q[Nat] means
    ( F.($1+1) is subsequence of F.$1
     & ||.F.($1+1).|| is bounded & (F.($1+1))#(x.($1+1)) is convergent
     & (F.($1+1))#(x.($1+1)) is subsequence of (F.$1)#(x.($1+1))
     & N.($1+1) is increasing sequence of NAT
     & F.($1+1) = (F.$1)*N.($1+1) );
Q0: Q[0] by X2,P0;
Q1: for n be Nat st Q[n] holds Q[n+1] by X2;
Q2: for n be Nat holds Q[n] from NAT_1:sch 2(Q0,Q1);
  take F,N;
  thus thesis by P0,X2,R0,Q2;
end;
