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 Th65:
for X be RealNormSpace, Sq be sequence of X st
 Sq is Cauchy_sequence_by_Norm holds
  ex N be increasing sequence of NAT st
   for i,j be Nat st j >= N.i holds
    ||. Sq.j - Sq.(N.i) .|| < 2 to_power (-i)
proof
   let X be RealNormSpace, Sq be sequence of X;
   assume A1: Sq is Cauchy_sequence_by_Norm;
   1 =2 to_power (-0) by POWER:24; then
   consider N0 be Nat such that
A2: for j,i be Nat st j >= N0 & i >= N0 holds
      ||. Sq.j - Sq.i .|| < 2 to_power (-0) by A1,RSSPACE3:8;
   reconsider N0 as Element of NAT by ORDINAL1:def 12;
   defpred P[set,set,set] means
    ex n,x,y be Nat st n=$1 & x=$2 & y=$3 &
     (  ( for j be Nat st j >= x holds
             ||.Sq.j -Sq.x.|| < 2 to_power (-n) ) implies x < y &
        ( for j be Nat st j >= y holds
             ||.Sq.j -Sq.y.|| < 2 to_power (-(n+1)) ) );
A3:for n being Nat,x being Element of NAT
   ex y being Element of NAT st P[n,x,y]
   proof
    let n be Nat, x be Element of NAT;
    now assume for j be Nat st j >= x
       holds ||.Sq.j -Sq.x.|| < 2 to_power (-n);
     0 < 2 to_power (-(n+1)) by POWER:34; then
     consider N2 be Nat such that
A4:  for j,i be Nat st j >= N2 & i >= N2 holds
        ||.Sq.j -Sq.i.|| < 2 to_power (-(n+1)) by A1,RSSPACE3:8;
     set y= max(x,N2)+1;
     take y;
     x <= max(x,N2) by XXREAL_0:25;
     hence x < y by NAT_1:13;
     N2 <= max(x,N2) by XXREAL_0:25; then
A5: N2 < y by NAT_1:13;
     thus for j be Nat st j >= y holds
              ||.Sq.j -Sq.y.|| < 2 to_power (-(n+1))
     proof
      let j be Nat;
      assume j >= y; then
      j >= N2 & y >=N2 by A5,XXREAL_0:2;
      hence thesis by A4;
     end;
    end;
    hence thesis;
   end;
   consider f being sequence of NAT such that
A6: f.0 = N0 &
    for n being Nat holds P[n,f.n,f.(n+1)] from RECDEF_1:sch 2(A3);
   defpred Q[Nat] means
    for j be Nat st j >= f.$1 holds
       ||. Sq.j - Sq.(f.$1) .|| < 2 to_power (-$1);
A7:Q[ 0] by A2,A6;
A8:now let i be Nat;
    assume A9: Q[i];
    ex n,x,y be Nat st n=i & x=f.i & y=f.(i+1) &
     (  (for j be Nat st j >= x holds
          ||.Sq.j -Sq.x.|| < 2 to_power (-n))
        implies
         x < y &
         (for j be Nat st j >= y holds
           ||.Sq.j -Sq.y.|| < 2 to_power (-(n+1)) )
     ) by A6;
    hence Q[i+1] by A9;
   end;
A10:for i be Nat holds Q[i] from NAT_1:sch 2(A7,A8);
   now let i be Nat;
    ex n,x,y be Nat st n=i & x=f.i & y=f.(i+1) &
     ( (for j be Nat st j >= x holds
          ||.Sq.j -Sq.x.|| < 2 to_power (-n))
       implies
        x < y &
        (for j be Nat st j >= y holds
           ||.Sq.j -Sq.y.|| < 2 to_power (-(n+1)))
     ) by A6;
    hence f.i < f.(i+1) by A10;
   end; then
   f is increasing;
   hence thesis by A10;
end;
