
theorem Th6:
  for X be RealBanachSpace, Y be RealNormSpace, vseq be sequence of
R_NormSpace_of_BoundedLinearOperators(X,Y), tseq be Function of X,Y st ( for x
be Point of X holds vseq#x is convergent & tseq.x = lim(vseq#x) ) holds tseq is
  Lipschitzian LinearOperator of X,Y & (for x be Point of X holds ||.tseq.x.||
  <=(lim_inf ||.vseq.|| ) * ||.x.|| ) & for ttseq be Point of
R_NormSpace_of_BoundedLinearOperators(X,Y) st ttseq = tseq holds ||.ttseq.|| <=
  lim_inf ||.vseq.||
proof
  let X be RealBanachSpace, Y be RealNormSpace, vseq be sequence of
  R_NormSpace_of_BoundedLinearOperators(X,Y), tseq be Function of X,Y;
  set T=rng vseq;
  set RNS=R_NormSpace_of_BoundedLinearOperators(X,Y);
  assume
A1: for x be Point of X holds vseq#x is convergent & tseq.x = lim(vseq#x );
A2: for x,y be Point of X holds tseq.(x+y)= tseq.x + tseq.y
  proof
    let x,y be Point of X;
A3: vseq#y is convergent & tseq.y = lim (vseq#y) by A1;
A4: tseq.(x+y) = lim (vseq#(x+y)) by A1;
    now
      let n be Nat;
      vseq.n is Lipschitzian LinearOperator of X,Y & (vseq#(x+y)).n=(vseq.n).(
      x+y) by Def2,LOPBAN_1:def 9;
      then
A5:   (vseq#(x+y)).n=(vseq.n).x + (vseq.n).y by VECTSP_1:def 20;
      (vseq.n).y = (vseq#y).n by Def2;
      hence (vseq#(x+y)).n=(vseq#x).n + (vseq#y).n by A5,Def2;
    end;
    then
A6: vseq#(x+y) = vseq#x + vseq#y by NORMSP_1:def 2;
    vseq#x is convergent & tseq.x = lim (vseq#x) by A1;
    hence thesis by A3,A6,A4,NORMSP_1:25;
  end;
A7: for x be Point of X ex K be Real st 0 <= K & for f be Point of
  RNS st f in T holds ||. f.x .|| <= K
  proof
    let x be Point of X;
    vseq#x is convergent by A1;
    then ||. vseq#x .|| is bounded by NORMSP_1:23,SEQ_2:13;
    then consider K be Real such that
A8: for n be Nat holds ||. vseq#x .||.n< K by SEQ_2:def 3;
A9: for f be Point of RNS st f in T holds ||. f.x .|| <= K
    proof
      let f be Point of RNS;
      assume f in T;
      then consider n be object such that
A10:  n in NAT and
A11:  f=vseq.n by FUNCT_2:11;
      reconsider n as Nat by A10;
      (vseq.n).x = (vseq#x).n by Def2;
      then ||. f.x .|| = ||. vseq#x .||.n by A11,NORMSP_0:def 4;
      hence thesis by A8;
    end;
    ||. vseq#x .||.0< K by A8;
    then ||. (vseq#x).0 .|| < K by NORMSP_0:def 4;
    then 0 <= K;
    hence thesis by A9;
  end;
  vseq in Funcs(NAT,the carrier of RNS) by FUNCT_2:8;
  then
  ex f0 being Function st vseq = f0 & dom f0 = NAT & rng f0 c= the carrier
  of RNS by FUNCT_2:def 2;
  then consider L be Real such that
A12: 0 <= L and
A13: for f be Point of RNS st f in T holds ||.f.|| <= L by A7,Th5;
A14: L + 0 < 1+ L by XREAL_1:8;
  for n be Nat holds |.||.vseq.||.n .| < (1+L)
  proof
    let n be Nat;
A15:    n in NAT by ORDINAL1:def 12;
    ||.vseq.n.|| <= L by A13,FUNCT_2:4,A15;
    then ||.vseq.||.n <= L by NORMSP_0:def 4;
    then
A16: ||.vseq.||.n <(1+L) by A14,XXREAL_0:2;
    0<=||.vseq.n.||;
    then 0<=||.vseq.||.n by NORMSP_0:def 4;
    hence thesis by A16,ABSVALUE:def 1;
  end;
  then
A17: ||.vseq.|| is bounded by A12,SEQ_2:3;
A18: for x be Point of X holds ||.tseq.x.|| <=( lim_inf ||.vseq.|| ) * ||.x .||
  proof
    let x be Point of X;
A19: ||.x.|| (#) ||.vseq .|| is bounded by A17,SEQM_3:37;
A20: for n be Nat holds ||.(vseq#x).n.|| <= ||.vseq.n.|| * ||.x .||
    proof
      let n be Nat;
      (vseq.n).x = (vseq#x).n & vseq.n is Lipschitzian LinearOperator of X,Y
      by Def2,LOPBAN_1:def 9;
      hence thesis by LOPBAN_1:32;
    end;
A21: for n be Nat holds ||. vseq#x .||.n <= (||.x.|| (#) ||.
    vseq .||).n
    proof
      let n be Nat;
A22:  ||.vseq.n.|| = ||.vseq.||.n by NORMSP_0:def 4;
      ||. vseq#x .||.n = ||.(vseq#x).n .|| & ||.(vseq#x).n .|| <= ||.vseq
      .n.|| * ||.x.|| by A20,NORMSP_0:def 4;
      hence thesis by A22,SEQ_1:9;
    end;
A23: lim_inf (||.x.|| (#) ||.vseq .||) = ( lim_inf ||.vseq.|| ) * ||.x.||
    by A17,Th1;
A24: vseq#x is convergent & tseq.x = lim(vseq#x) by A1;
    then ||. vseq#x .|| is convergent by LOPBAN_1:20;
    then
A25: lim ||. vseq#x .|| = lim_inf ||. vseq#x .|| by RINFSUP1:89;
    ||. vseq#x .|| is bounded by A24,LOPBAN_1:20,SEQ_2:13;
    then lim_inf ||. vseq#x .|| <=lim_inf (||.x.|| (#) ||.vseq .||) by A19,A21,
RINFSUP1:91;
    hence thesis by A24,A23,A25,LOPBAN_1:20;
  end;
  now
    let s be Real;
    assume
A26: 0<s;
    for k be Nat holds 0-s < ||.vseq.||.(0+k)
    proof
      let k be Nat;
      ||.vseq.k.||=||.vseq.||.k by NORMSP_0:def 4;
      then 0<=||.vseq.||.k;
      hence thesis by A26;
    end;
    hence ex n be Nat st for k be Nat holds 0-s<||.vseq.||.(n+k);
  end;
  then
A27: 0 <= lim_inf||.vseq.|| by A17,RINFSUP1:82;
A28: for x be Point of X, r be Real holds tseq.(r*x)= r*tseq.x
  proof
    let x be Point of X, r be Real;
A29: tseq.x = lim(vseq#x) by A1;
A30: now
      let n be Nat;
      vseq.n is Lipschitzian LinearOperator of X,Y & (vseq#(r*x)).n=(vseq.n).(
      r*x) by Def2,LOPBAN_1:def 9;
      then (vseq#(r*x)).n=r*(vseq.n).x by LOPBAN_1:def 5;
      hence (vseq#(r*x)).n=r*(vseq#x).n by Def2;
    end;
    tseq.(r*x) = lim (vseq#(r*x)) by A1;
    then tseq.(r*x) = lim (r*(vseq#x)) by A30,NORMSP_1:def 5;
    hence thesis by A1,A29,NORMSP_1:28;
  end;
  then reconsider tseq1 = tseq as Lipschitzian LinearOperator of X,Y
  by A2,A18,A27,LOPBAN_1:def 5,def 8,VECTSP_1:def 20;
  for ttseq be Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st
  ttseq = tseq holds ||.ttseq.|| <=( lim_inf ||.vseq.|| )
  proof
    for k be Real st k in {||.tseq.x1.|| where x1 is Point of X :
    ||.x1.|| <= 1 } holds k <= ( lim_inf ||.vseq.|| )
    proof
      let k be Real;
      assume k in {||.tseq.x1.|| where x1 is Point of X : ||.x1.|| <= 1 };
      then consider x be Point of X such that
A31:  k=||.tseq.x.|| & ||.x.|| <= 1;
      k <= (lim_inf ||.vseq.||) * ||.x.|| & (lim_inf ||.vseq.||) * ||.x
      .|| <= lim_inf ||.vseq.|| by A18,A27,A31,XREAL_1:153;
      hence thesis by XXREAL_0:2;
    end;
    then
A32: upper_bound PreNorms(tseq1) <=lim_inf ||.vseq.|| by SEQ_4:45;
    let ttseq be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume ttseq=tseq;
    hence thesis by A32,LOPBAN_1:30;
  end;
  hence thesis by A2,A28,A18,A27,LOPBAN_1:def 5,def 8,VECTSP_1:def 20;
end;
