reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th42:
  for X, Y be RealNormSpace st Y is complete for seq be sequence
of R_NormSpace_of_BoundedLinearOperators(X,Y) st seq is Cauchy_sequence_by_Norm
  holds seq is convergent
proof
  let X, Y be RealNormSpace such that
A1: Y is complete;
  let vseq be sequence of R_NormSpace_of_BoundedLinearOperators(X,Y) such that
A2: vseq is Cauchy_sequence_by_Norm;
  defpred P[set, set] means ex xseq be sequence of Y st
   (for n be Nat holds xseq.n=modetrans((vseq.n),X,Y).$1) &
     xseq is convergent & $2= lim xseq;
A3: for x be Element of X ex y be Element of Y st P[x,y]
  proof
    let x be Element of X;
    deffunc F(Nat) = modetrans((vseq.$1),X,Y).x;
    consider xseq be sequence of Y such that
A4: for n be Element of NAT holds xseq.n = F(n) from FUNCT_2:sch 4;
A5: for n be Nat holds xseq.n = F(n)
     proof let n be Nat;
      n in NAT by ORDINAL1:def 12;
      hence thesis by A4;
     end;
    take lim xseq;
A6: for m,k be Nat holds ||.xseq.m-xseq.k.|| <= ||.vseq.m -
    vseq.k.|| * ||.x.||
    proof
      let m,k be Nat;
      reconsider h1=vseq.m-vseq.k as Lipschitzian LinearOperator of X,Y
      by Def9;
      k in NAT by ORDINAL1:def 12;
      then
A7:   xseq.k =modetrans((vseq.k),X,Y).x by A4;
      vseq.m is Lipschitzian LinearOperator of X,Y by Def9;
      then
A8:   modetrans((vseq.m),X,Y)=vseq.m by Th29;
      vseq.k is Lipschitzian LinearOperator of X,Y by Def9;
      then
A9:   modetrans((vseq.k),X,Y)=vseq.k by Th29;
      m in NAT by ORDINAL1:def 12;
      then
      xseq.m =modetrans((vseq.m),X,Y).x by A4;
      then xseq.m - xseq.k = h1.x by A8,A9,A7,Th40;
      hence thesis by Th32;
    end;
    now
      let e be Real such that
A10:   e > 0;
      now
        per cases;
        case
A11:      x=0.X;
          reconsider k=0 as Nat;
          take k;
          thus for n, m be Nat st n >= k & m >= k holds ||.xseq.n -
          xseq.m.|| < e
          proof
            let n, m be Nat such that
            n >= k and
            m >= k;
      m in NAT by ORDINAL1:def 12;
      then
A12:        xseq.m=modetrans((vseq.m),X,Y).x by A4
              .=modetrans((vseq.m),X,Y).(0*x) by A11
              .=0*modetrans((vseq.m),X,Y).x by Def5
              .=0.Y by RLVECT_1:10;
      n in NAT by ORDINAL1:def 12;
      then
            xseq.n=modetrans((vseq.n),X,Y).x by A4
              .=modetrans((vseq.n),X,Y).(0*x) by A11
              .=0*modetrans((vseq.n),X,Y).x by Def5
              .=0.Y by RLVECT_1:10;
            then ||.xseq.n -xseq.m.|| = ||.0.Y.|| by A12
              .=0;
            hence thesis by A10;
          end;
        end;
        case
          x <>0.X;
          then
A13:      ||.x.|| <> 0 by NORMSP_0:def 5;
          then
A14:      ||.x.|| > 0;
          then e/||.x.||>0 by A10,XREAL_1:139;
          then consider k be Nat such that
A15:      for n, m be Nat st n >= k & m >= k holds ||.(
          vseq.n) - (vseq.m).|| < e/||.x.|| by A2,RSSPACE3:8;
          take k;
          thus for n, m be Nat st n >= k & m >= k holds ||.xseq.n-
          xseq.m.|| < e
          proof
            let n,m be Nat such that
A16:        n >=k and
A17:        m >= k;
            ||.(vseq.n) - (vseq.m).|| < e/||.x.|| by A15,A16,A17;
            then
A18:        ||.(vseq.n) - (vseq.m).|| * ||.x.|| < e/||.x.|| * ||.x.|| by A14,
XREAL_1:68;
A19:        e/||.x.|| * ||.x.|| = e*||.x.||"* ||.x.|| by XCMPLX_0:def 9
              .= e*(||.x.||"* ||.x.||)
              .= e*1 by A13,XCMPLX_0:def 7
              .=e;
            ||.xseq.n-xseq.m.|| <= ||.(vseq.n) - (vseq.m).|| * ||.x.|| by A6;
            hence thesis by A18,A19,XXREAL_0:2;
          end;
        end;
      end;
      hence ex k be Nat st for n, m be Nat st n >= k & m
      >= k holds ||.xseq.n -xseq.m.|| < e;
    end;
    then xseq is Cauchy_sequence_by_Norm by RSSPACE3:8;
    then xseq is convergent by A1;
    hence thesis by A5;
  end;
  consider f be Function of the carrier of X,the carrier of Y such that
A20: for x be Element of X holds P[x,f.x] from FUNCT_2:sch 3(A3);
  reconsider tseq=f as Function of X,Y;
A21: now
    let x,y be VECTOR of X;
    consider xseq be sequence of Y such that
A22: for n be Nat holds xseq.n=modetrans((vseq.n),X,Y).x and
A23: xseq is convergent and
A24: tseq.x = lim xseq by A20;
    consider zseq be sequence of Y such that
A25: for n be Nat holds zseq.n=modetrans((vseq.n),X,Y).(x+y ) and
    zseq is convergent and
A26: tseq.(x+y) = lim zseq by A20;
    consider yseq be sequence of Y such that
A27: for n be Nat holds yseq.n=modetrans((vseq.n),X,Y).y and
A28: yseq is convergent and
A29: tseq.y = lim yseq by A20;
    now
      let n be Nat;
      thus zseq.n=modetrans((vseq.n),X,Y).(x+y) by A25
        .= modetrans((vseq.n),X,Y).x+modetrans((vseq.n),X,Y).y
         by VECTSP_1:def 20
        .= xseq.n + modetrans((vseq.n),X,Y).y by A22
        .= xseq.n +yseq.n by A27;
    end;
    then zseq=xseq+yseq by NORMSP_1:def 2;
    hence tseq.(x+y)=tseq.x+tseq.y by A23,A24,A28,A29,A26,NORMSP_1:25;
  end;
  now
    let x be VECTOR of X;
    let a be Real;
    consider xseq be sequence of Y such that
A30: for n be Nat holds xseq.n=modetrans((vseq.n),X,Y).x and
A31: xseq is convergent and
A32: tseq.x = lim xseq by A20;
    consider zseq be sequence of Y such that
A33: for n be Nat holds zseq.n=modetrans((vseq.n),X,Y).(a*x ) and
    zseq is convergent and
A34: tseq.(a*x) = lim zseq by A20;
    now
      let n be Nat;
      thus zseq.n=modetrans((vseq.n),X,Y).(a*x) by A33
        .= a*modetrans((vseq.n),X,Y).x by Def5
        .= a*xseq.n by A30;
    end;
    then zseq=a*xseq by NORMSP_1:def 5;
    hence tseq.(a*x)=a*tseq.x by A31,A32,A34,NORMSP_1:28;
  end;
  then reconsider tseq as LinearOperator of X,Y by A21,Def5,VECTSP_1:def 20;
  now
    let e1 be Real such that
A35: e1 >0;
    reconsider e =e1 as Real;
    consider k be Nat such that
A36: for n, m be Nat st n >= k & m >= k holds ||.(vseq.n) -
    (vseq.m).|| < e by A2,A35,RSSPACE3:8;
     reconsider k as Nat;
    take k;
    now
      let m be Nat;
      assume m >= k;
      then
A37:  ||.(vseq.m) - (vseq.k).|| <e by A36;
A38:  ||.vseq.m.||= ||.vseq.||.m by NORMSP_0:def 4;
A39:  ||.vseq.k.||= ||.vseq.||.k by NORMSP_0:def 4;
      |. ||.vseq.m.||- ||.vseq.k.|| .| <= ||.(vseq.m) - (vseq.k).|| by
NORMSP_1:9;
      hence |. ||.vseq.||.m - ||.vseq.||.k .| <e1 by A39,A38,A37,XXREAL_0:2;
    end;
    hence
    for m be Nat st m >= k holds |.||.vseq.||.m - ||.vseq.||
    .k .| < e1;
  end;
  then
A40: ||.vseq.|| is convergent by SEQ_4:41;
A41: tseq is Lipschitzian
  proof
    take lim (||.vseq.|| );
A42: now
      let x be VECTOR of X;
      consider xseq be sequence of Y such that
A43:  for n be Nat holds xseq.n=modetrans((vseq.n),X,Y).x and
A44:  xseq is convergent and
A45:  tseq.x = lim xseq by A20;
A46:  ||.tseq.x.|| = lim ||.xseq.|| by A44,A45,Th20;
A47:  for m be Nat holds ||.xseq.m.|| <= ||.vseq.m.|| * ||.x .||
      proof
        let m be Nat;
A48:    xseq.m =modetrans((vseq.m),X,Y).x by A43;
        vseq.m is Lipschitzian LinearOperator of X,Y by Def9;
        hence thesis by A48,Th29,Th32;
      end;
A49:  for n be Nat holds ||.xseq.||.n <= ( ||.x.||(#)||.vseq .||).n
      proof
        let n be Nat;
A50:    ||.xseq.||.n = ||.(xseq.n).|| by NORMSP_0:def 4;
A51:    ||.vseq.n.|| = ||.vseq.||.n by NORMSP_0:def 4;
        ||.(xseq.n).|| <= ||.vseq.n.|| * ||.x.|| by A47;
        hence thesis by A50,A51,SEQ_1:9;
      end;
A52:  ||.x.||(#)||.vseq.|| is convergent by A40;
A53:  lim ( ||.x.||(#)||.vseq.|| ) = lim (||.vseq.|| )* ||.x.|| by A40,SEQ_2:8;
      ||.xseq.|| is convergent by A44,A45,Th20;
      hence ||.tseq.x.|| <= lim (||.vseq.|| )* ||.x.|| by A46,A49,A52,A53,
SEQ_2:18;
    end;
    now
      let n be Nat;
      ||.vseq.n.|| >=0;
      hence ||.vseq.||.n >=0 by NORMSP_0:def 4;
    end;
    hence thesis by A40,A42,SEQ_2:17;
  end;
A54: for e be Real
    st e > 0 ex k be Nat st for n be Nat st n >= k
    holds for x be VECTOR of X holds ||.modetrans((vseq.n),X,Y).x -
  tseq.x.|| <= e* ||.x.||
  proof
    let e be Real;
    assume e > 0;
    then consider k be Nat such that
A55: for n, m be Nat st n >= k & m >= k holds ||.(vseq.n) -
    (vseq.m).|| < e by A2,RSSPACE3:8;
    take k;
    now
      let n be Nat such that
A56:  n >= k;
      now
        let x be VECTOR of X;
        consider xseq be sequence of Y such that
A57:    for n be Nat holds xseq.n=modetrans((vseq.n),X,Y). x and
A58:    xseq is convergent and
A59:    tseq.x = lim xseq by A20;
A60:    for m,k be Nat holds ||.xseq.m-xseq.k.|| <= ||.vseq.m
        - vseq.k.|| * ||.x.||
        proof
          let m,k be Nat;
          reconsider h1=vseq.m-vseq.k as Lipschitzian LinearOperator of X,Y
          by Def9;
A61:      xseq.k =modetrans((vseq.k),X,Y).x by A57;
          vseq.m is Lipschitzian LinearOperator of X,Y by Def9;
          then
A62:      modetrans((vseq.m),X,Y)=vseq.m by Th29;
          vseq.k is Lipschitzian LinearOperator of X,Y by Def9;
          then
A63:      modetrans((vseq.k),X,Y)=vseq.k by Th29;
          xseq.m =modetrans((vseq.m),X,Y).x by A57;
          then xseq.m - xseq.k =h1.x by A62,A63,A61,Th40;
          hence thesis by Th32;
        end;
A64:    for m be Nat st m >=k holds ||.xseq.n-xseq.m.|| <= e *
        ||.x.||
        proof
          let m be Nat;
          assume m >=k;
          then
A65:      ||.vseq.n - vseq.m.|| <e by A55,A56;
A66:      ||.xseq.n-xseq.m.|| <= ||.vseq.n - vseq.m.|| * ||.x.|| by A60;
          ||.vseq.n - vseq.m.|| * ||.x.|| <= e* ||.x.|| by A65,XREAL_1:64;
          hence thesis by A66,XXREAL_0:2;
        end;
        ||.xseq.n-tseq.x.|| <= e * ||.x.||
        proof
          deffunc F(Nat) = ||.xseq.$1 - xseq.n.||;
          consider rseq be Real_Sequence such that
A67:      for m be Nat holds rseq.m = F(m) from SEQ_1:sch
          1;
          now
            let x be object;
            assume x in NAT;
            then reconsider k=x as Nat;
            thus rseq.x = ||.xseq.k - xseq.n.|| by A67
              .= ||.(xseq - xseq.n).k.|| by NORMSP_1:def 4
              .= ||.(xseq - xseq.n).||.x by NORMSP_0:def 4;
          end;
          then
A68:      rseq = ||.xseq - xseq.n.|| by FUNCT_2:12;
A69:      xseq - xseq.n is convergent by A58,NORMSP_1:21;
          lim (xseq-xseq.n)= tseq.x - xseq.n by A58,A59,NORMSP_1:27;
          then
A70:      lim rseq = ||.tseq.x-xseq.n.|| by A69,A68,Th41;
          for m be Nat st m >= k holds rseq.m <= e * ||.x.||
          proof
            let m be Nat such that
A71:        m >=k;
            rseq.m = ||.xseq.m-xseq.n.|| by A67
              .= ||.xseq.n-xseq.m.|| by NORMSP_1:7;
            hence thesis by A64,A71;
          end;
          then lim rseq <= e * ||.x.|| by A69,A68,Lm3,Th41;
          hence thesis by A70,NORMSP_1:7;
        end;
        hence ||.modetrans((vseq.n),X,Y).x - tseq.x.|| <= e* ||.x.|| by A57;
      end;
      hence
      for x be VECTOR of X holds ||.modetrans((vseq.n),X,Y).x - tseq.x.||
      <= e* ||.x.||;
    end;
    hence thesis;
  end;
  reconsider tseq as Lipschitzian LinearOperator of X,Y by A41;
  reconsider tv=tseq as Point of R_NormSpace_of_BoundedLinearOperators(X,Y) by
Def9;
A72: for e be Real st e > 0
  ex k be Nat st for n be Nat st n >= k holds ||.vseq.n - tv.|| <= e
  proof
    let e be Real such that
A73: e > 0;
    consider k be Nat such that
A74: for n be Nat st n >= k holds for x be VECTOR of X
    holds ||.modetrans((vseq.n),X,Y).x - tseq.x.|| <= e* ||.x.|| by A54,A73;
    now
      set g1=tseq;
      let n be Nat such that
A75:  n >= k;
      reconsider h1=vseq.n-tv as Lipschitzian LinearOperator of X,Y by Def9;
      set f1=modetrans((vseq.n),X,Y);
A76:  now
        let t be VECTOR of X;
        assume ||.t.|| <= 1;
        then
A77:    e*||.t.|| <= e*1 by A73,XREAL_1:64;
A78:    ||.f1.t-g1.t.|| <=e* ||.t.|| by A74,A75;
        vseq.n is Lipschitzian LinearOperator of X,Y by Def9;
        then modetrans((vseq.n),X,Y)=vseq.n by Th29;
        then ||.h1.t.||= ||.f1.t-g1.t.|| by Th40;
        hence ||.h1.t.|| <=e by A78,A77,XXREAL_0:2;
      end;
A79:  now
        let r be Real;
        assume r in PreNorms(h1);
        then ex t be VECTOR of X st r=||.h1.t.|| & ||.t.|| <= 1;
        hence r <=e by A76;
      end;
A80:  (for s be Real st s in PreNorms(h1) holds s <= e) implies
      upper_bound PreNorms(h1) <=e by SEQ_4:45;
      BoundedLinearOperatorsNorm(X,Y).(vseq.n-tv) = upper_bound PreNorms(h1)
      by Th30;
      hence ||.vseq.n-tv.|| <=e by A79,A80;
    end;
    hence thesis;
  end;
  for e be Real
    st e > 0 ex m be Nat st for n be Nat st n >= m
         holds ||.(vseq.n) - tv.|| < e
  proof
    let e be Real such that
A81: e > 0;
    consider m be Nat such that
A82: for n be Nat st n >= m holds ||.(vseq.n) - tv.|| <= e
    /2 by A72,A81,XREAL_1:215;
A83: e/2<e by A81,XREAL_1:216;
    now
      let n be Nat;
      assume n >= m;
      then ||.(vseq.n) - tv.|| <= e/2 by A82;
      hence ||.(vseq.n) - tv.|| < e by A83,XXREAL_0:2;
    end;
    hence thesis;
  end;
  hence thesis by NORMSP_1:def 6;
end;
