
theorem
  for X,Y be RealBanachSpace, X0 be Subset of LinearTopSpaceNorm(X),
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 ex tseq be Point of
R_NormSpace_of_BoundedLinearOperators(X,Y) st ( for x be Point of X holds vseq#
  x is convergent & tseq.x=lim (vseq#x) & ||.tseq.x.|| <= lim_inf ||.vseq.|| *
  ||.x.|| ) & ||.tseq.|| <= lim_inf ||.vseq.||
proof
  let X,Y be RealBanachSpace, X0 be Subset of LinearTopSpaceNorm(X), vseq be
  sequence of R_NormSpace_of_BoundedLinearOperators(X,Y);
  assume
A1: X0 is dense;
  deffunc F(Point of X) = lim(vseq#$1);
  assume
A2: for x be Point of X st x in X0 holds vseq#x is convergent;
  consider tseq be Function of X,Y such that
A3: for x be Point of X holds tseq.x=F(x) from FUNCT_2:sch 4;
  assume for x be Point of X ex K be Real st 0<=K &
   for n be Nat holds ||.(vseq#x).n.|| <= K;
  then
A4: for x be Point of X holds vseq#x is convergent by A1,A2,Th7;
  then reconsider tseq as Lipschitzian LinearOperator of X,Y by A3,Th6;
  reconsider tseq as Point of R_NormSpace_of_BoundedLinearOperators(X,Y) by
LOPBAN_1:def 9;
  take tseq;
  thus thesis by A4,A3,Th6;
end;
