
theorem Th7:
  for X be RealBanachSpace, X0 be Subset of LinearTopSpaceNorm(X),
Y be RealBanachSpace, vseq be sequence of R_NormSpace_of_BoundedLinearOperators
  (X,Y) st X0 is dense & (for x be Point of X st x in X0 holds vseq#x is
  convergent) & (for x be Point of X ex K be Real st 0<=K & for n be
Nat holds ||.(vseq#x).n.|| <=K) holds for x be Point of X holds vseq
  #x is convergent
proof
  let X be RealBanachSpace, X0 be Subset of LinearTopSpaceNorm(X), Y be
RealBanachSpace, vseq be sequence of R_NormSpace_of_BoundedLinearOperators(X,Y)
  ;
  assume
A1: X0 is dense;
  set T=rng vseq;
  assume
A2: for x be Point of X st x in X0 holds vseq#x is convergent;
  vseq in Funcs(NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(X
  ,Y) ) by FUNCT_2:8;
  then
  ex f0 being Function st vseq = f0 & dom f0 = NAT & rng f0 c= the carrier
  of R_NormSpace_of_BoundedLinearOperators(X,Y) by FUNCT_2:def 2;
  then reconsider T as Subset of R_NormSpace_of_BoundedLinearOperators(X,Y);
  assume
A3: for x be Point of X ex K be Real st 0<=K &
   for n be Nat holds ||.(vseq#x).n.|| <=K;
  for x be Point of X ex K be Real st 0 <= K & for f be Point of
  R_NormSpace_of_BoundedLinearOperators(X,Y) st f in T holds ||. f.x .|| <= K
  proof
    let x be Point of X;
    consider K be Real such that
A4: 0<= K and
A5: for n be Nat holds ||.(vseq#x).n.|| <=K by A3;
    take K;
    now
      let f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
      assume f in T;
      then consider n be object such that
A6:   n in NAT and
A7:   f=vseq.n by FUNCT_2:11;
      reconsider n as Nat by A6;
      ||. f.x .|| = ||.(vseq#x).n.|| by A7,Def2;
      hence ||. f.x .|| <= K by A5;
    end;
    hence thesis by A4;
  end;
  then consider L be Real such that
A8: 0 <= L and
A9: for f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st f
  in T holds ||.f.|| <= L by Th5;
  set M=1+L;
A10: L+0<M by XREAL_1:8;
A11: for f be Lipschitzian LinearOperator of X,Y st f in T for x,y be Point
  of X holds ||. f.x -f.y.|| <= M*||.x-y.||
  proof
    let f be Lipschitzian LinearOperator of X,Y;
    reconsider f1=f as Point of R_NormSpace_of_BoundedLinearOperators(X,Y) by
LOPBAN_1:def 9;
    assume f in T;
    then ||.f1.|| <= L by A9;
    then
A12: ||.f1.|| < M by A10,XXREAL_0:2;
    let x,y be Point of X;
    ||. f.x -f.y.|| = ||. f.x +(-1)*(f.y).|| by RLVECT_1:16;
    then ||. f.x -f.y.|| = ||. f.x +(f.((-1)*y)).|| by LOPBAN_1:def 5;
    then ||. f.x -f.y.|| = ||. f.(x +(-1)*y).|| by VECTSP_1:def 20;
    then
A13: ||. f.x -f.y.|| = ||. f.(x -y).||by RLVECT_1:16;
    ||. f.(x -y).|| <=||.f1.|| * ||.x-y.|| & ||.f1.||* ||.x-y.|| <= M*
    ||.x-y.|| by A12,LOPBAN_1:32,XREAL_1:64;
    hence thesis by A13,XXREAL_0:2;
  end;
  hereby
    let x be Point of X;
    for TK1 be Real st TK1 > 0 ex n0 be Nat st for n,m be
    Nat st n >= n0 & m >= n0 holds ||.(vseq#x).n -(vseq#x).m.||< TK1
    proof
      let TK1 be Real such that
A14:  TK1 > 0;
A15:  0<TK1/3 by A14,XREAL_1:222;
      set V = {z where z is Point of X : ||.x-z.|| <TK1/(3*M) };
      V c= the carrier of X
      proof
        let s be object;
        assume s in V;
        then ex z be Point of X st s=z & ||.x-z.|| <TK1/(3*M);
        hence thesis;
      end;
      then reconsider V as Subset of LinearTopSpaceNorm(X) by NORMSP_2:def 4;
      reconsider TKK=TK1 as Real;
      0 < (3*M) by A8,XREAL_1:129;
      then 0 < TK1/(3*M) by A14,XREAL_1:139;
      then ||.x-x.|| <TKK/(3*M) by NORMSP_1:6;
      then V is open & x in V by NORMSP_2:23;
      then X0 meets V by A1,TOPS_1:45;
      then consider s be object such that
A16:  s in X0 and
A17:  s in V by XBOOLE_0:3;
      consider y be Point of X such that
A18:  s=y and
A19:  ||.x-y.|| < TK1/(3*M) by A17;
      for s be Real st 0<s ex n1 be Nat st
     for m1 be Nat st n1<=m1 holds ||.(vseq#y).m1 -(vseq#y).n1.||<s
        by A2,A16,A18,
LOPBAN_3:4;
      then vseq#y is Cauchy_sequence_by_Norm by LOPBAN_3:5;
      then consider n0 be Nat such that
A20:  for n, m be Nat st n >= n0 & m >= n0 holds ||.((vseq
      #y).n) - ((vseq#y).m).|| < TK1/3 by A15,RSSPACE3:8;
      take n0;
      for n, m be Nat st n >= n0 & m >= n0 holds ||.(vseq#x).n
      -(vseq#x).m.||< TK1
      proof
        let n,m be Nat;
A21:  m in NAT by ORDINAL1:def 12;
A22:  n in NAT by ORDINAL1:def 12;
        reconsider f = vseq.n as Lipschitzian LinearOperator of X,Y by
LOPBAN_1:def 9;
        reconsider g =vseq.m as Lipschitzian LinearOperator of X,Y
        by LOPBAN_1:def 9;
        ||. (vseq#x).n - (vseq#y).m .|| <= ||. (vseq#x).n - (vseq#y).n
        .|| + ||. (vseq#y).n - (vseq#y).m .|| by NORMSP_1:10;
        then
A23:    ||. (vseq#x).n - (vseq#y).m .|| + ||. (vseq#y).m - (vseq#x ).m
.|| <= ||. (vseq#x).n - (vseq#y).n .|| + ||. (vseq#y).n - (vseq#y).m .|| + ||.
        (vseq#y).m - (vseq#x).m .|| by XREAL_1:6;
        assume n >= n0 & m >= n0;
        then ||. (vseq#y).n - (vseq#y).m .|| < TK1/3 by A20;
        then
        ||. (vseq#x).n - (vseq#y).n .|| + ||. (vseq#y).n - (vseq#y).m .||
        < ||. (vseq#x).n - (vseq#y).n .||+ TK1/3 by XREAL_1:8;
        then
A24:    ||. (vseq#x).n - (vseq#y).n .||+ ||. (vseq#y).n - (vseq#y) .m .||
+ ||. (vseq#y).m - (vseq#x).m .|| < ||. (vseq#x).n - (vseq#y).n .||+ TK1/3 +
        ||. (vseq#y).m - (vseq#x).m .|| by XREAL_1:8;
        ||. (vseq#x).m-(vseq#y).m .|| = ||. (vseq.m).x-(vseq#y).m .|| by Def2;
        then ||. (vseq#x).m-(vseq#y).m .|| = ||. g.x-g.y.|| by Def2;
        then
A25:    ||. (vseq#x).m-(vseq#y).m .|| <= M*||.x-y.|| by A11,FUNCT_2:4,A21;
        M*||.x-y.||<M* (TK1 / (3*M)) by A8,A19,XREAL_1:68;
        then M*||.x-y.||< TK1/3 by A8,XCMPLX_1:92;
        then ||. (vseq#x).m-(vseq#y).m .|| < TK1/3 by A25,XXREAL_0:2;
        then ||. (vseq#y).m-(vseq#x).m .||< TK1/3 by NORMSP_1:7;
        then
A26:    TK1/3 + TK1/3 + ||. (vseq#y).m-(vseq#x).m .|| < TK1/3 + TK1/3 +
        TK1/3 by XREAL_1:8;
        ||. (vseq#x).n-(vseq#y).n .|| = ||. (vseq.n).x-(vseq#y).n .|| by Def2;
        then ||. (vseq#x).n-(vseq#y).n .|| = ||.f.x-f.y.|| by Def2;
        then
A27:    ||. (vseq#x).n-(vseq#y).n .|| <= M*||.x-y.|| by A11,FUNCT_2:4,A22;
        ||. (vseq#x).n - (vseq#x).m .|| <= ||. (vseq#x).n - (vseq#y).m
        .|| + ||. (vseq#y).m - (vseq#x).m .|| by NORMSP_1:10;
        then
        ||. (vseq#x).n -(vseq#x).m .|| <= ||. (vseq#x).n - (vseq#y).n .||
+ ||. (vseq#y).n - (vseq#y).m .|| + ||. (vseq#y).m - (vseq#x).m .|| by A23,
XXREAL_0:2;
        then
A28:    ||. (vseq#x).n -(vseq#x).m .|| < ||. (vseq#x).n - (vseq#y).n .||+
        TK1/3 + ||. (vseq#y).m - (vseq#x).m .|| by A24,XXREAL_0:2;
        M*||.x-y.||<M* (TK1 / (3*M)) by A8,A19,XREAL_1:68;
        then M*||.x-y.||< TK1/3 by A8,XCMPLX_1:92;
        then ||. (vseq#x).n - (vseq#y).n.|| < TK1/3 by A27,XXREAL_0:2;
        then ||. (vseq#x).n-(vseq#y).n .||+ TK1/3 < TK1/3+TK1/3 by XREAL_1:8;
        then
        ||. (vseq#x).n - (vseq#y).n .||+ TK1/3 + ||. (vseq#y).m - (vseq#x
).m .|| < TK1/3 + TK1/3 + ||. (vseq#y).m - (vseq#x).m .|| by XREAL_1:8;
        then
        ||. (vseq#x).n-(vseq#y).n .|| + TK1/3 + ||. (vseq#y).m-(vseq#x).m
        .|| < TK1/3 + TK1/3 + TK1/3 by A26,XXREAL_0:2;
        hence thesis by A28,XXREAL_0:2;
      end;
      hence thesis;
    end;
    then vseq#x is Cauchy_sequence_by_Norm by RSSPACE3:8;
    hence vseq#x is convergent by LOPBAN_1:def 15;
  end;
end;
