reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);

theorem Th34:
  for x0 be Point of S for N being Neighbourhood of x0 st f
is_differentiable_in x0 & N c= dom f holds
for h be (0.S)-convergent sequence of S st h is non-zero
 holds
   for c st rng c = {x0} & rng (h+c) c= N holds (f/*(h+c) - f/*c) is convergent
  & lim (f/*(h+c) - f/*c) = 0.T
proof
  let x0 be Point of S;
  let N be Neighbourhood of x0;
  assume that
A1: f is_differentiable_in x0 and
A2: N c= dom f;
  let h be (0.S)-convergent sequence of S;
  assume A3: h is non-zero;
  let c such that
A4: rng c = {x0} and
A5: rng (h+c) c= N;
  consider N1 be Neighbourhood of x0 such that
  N1 c= dom f and
A6: ex L,R st for x be Point of S st x in N1 holds f/.x - f/.x0 = L.(x-
  x0) + R/.(x-x0) by A1;
  consider N2 be Neighbourhood of x0 such that
A7: N2 c= N and
A8: N2 c= N1 by Th1;
  consider L,R such that
A9: for x be Point of S st x in N1 holds f/.x - f/.x0 = L.(x-x0) + R/.(x
  -x0) by A6;
  consider g be Real such that
A10: 0<g and
A11: {y where y is Point of S: ||.y-x0.|| < g} c= N2 by NFCONT_1:def 1;
A12: x0 in N2 by NFCONT_1:4;
  ex n st rng (c^\n) c= N2 & rng ((h+c)^\n) c= N2
  proof
    c.0 in rng c by NFCONT_1:6;
    then c.0=x0 by A4,TARSKI:def 1;
    then
A13: lim c = x0 by Th18;
A14: c is convergent & h is convergent by Def4,Th18;
    then
A15: h + c is convergent by NORMSP_1:19;
    lim h = 0.S by Def4;
    then lim (h + c) = 0.S + x0 by A13,A14,NORMSP_1:25
      .= x0 by RLVECT_1:4;
    then consider n being Nat such that
A16: for m being Nat st n<=m holds ||.(h+c).m-x0.||<g by A10,A15,
NORMSP_1:def 7;
     reconsider n as Element of NAT by ORDINAL1:def 12;
    take n;
A17: rng (c^\n) = {x0} by A4,VALUED_0:26;
    thus rng (c^\n) c= N2
    by A12,A17,TARSKI:def 1;
    let y be object;
    assume y in rng ((h+c)^\n);
    then consider m be Nat such that
A18: y = ((h+c)^\n).m by NFCONT_1:6;
    reconsider y1=y as Point of S by A18;
    n + 0 <= n+m by XREAL_1:7;
    then ||.(h+c).(n+m)-x0.||<g by A16;
    then ||.((h+c)^\n).m - x0 .|| < g by NAT_1:def 3;
    then y1 in {z where z is Point of S: ||.z-x0.|| < g} by A18;
    hence thesis by A11;
  end;
  then consider n such that
  rng (c^\n) c= N2 and
A19: rng ((h+c)^\n) c= N2;
A20: lim (h^\n) = 0.S by Def4;
A21: for k holds f/.(((h+c)^\n).k) - f/.((c^\n).k) = L.((h^\n).k) + R/.((h^\
  n).k)
  proof
    let k;
    ((h+c)^\n).k in rng ((h+c)^\n) by NFCONT_1:6;
    then
A22: ((h+c)^\n).k in N2 by A19;
    (c^\n).k in rng (c^\n) & rng (c^\n) = rng c by NFCONT_1:6,VALUED_0:26;
    then
A23: (c^\n).k = x0 by A4,TARSKI:def 1;
    ((h+c)^\n).k - (c^\n).k = (h+c).(k+n) - (c^\n).k by NAT_1:def 3
      .=h.(k+n)+ c.(k+n) - (c^\n).k by NORMSP_1:def 2
      .=(h^\n).k+ c.(k+n) - (c^\n).k by NAT_1:def 3
      .=(h^\n).k+ (c^\n).k- (c^\n).k by NAT_1:def 3
      .= (h^\n).k + ((c^\n).k - (c^\n).k) by RLVECT_1:def 3
      .= (h^\n).k + 0.S by RLVECT_1:15
      .= (h^\n).k by RLVECT_1:4;
    hence thesis by A9,A8,A22,A23;
  end;
  R_NormSpace_of_BoundedLinearOperators(S,T) = NORMSTR (#
    BoundedLinearOperators(S,T), Zero_(BoundedLinearOperators(S,T),
    R_VectorSpace_of_LinearOperators(S,T)), Add_(BoundedLinearOperators(S,T),
    R_VectorSpace_of_LinearOperators(S,T)), Mult_(BoundedLinearOperators(S,T),
R_VectorSpace_of_LinearOperators(S,T)), BoundedLinearOperatorsNorm(S,T) #) by
LOPBAN_1:def 14;
  then reconsider L as Element of BoundedLinearOperators(S,T);
  reconsider LP=modetrans(L,S,T) as PartFunc of S,T;
A24: h^\n is non-zero by A3,Th17;
then
A25: lim (R/*(h^\n)) = 0.T by Th24;
A26: rng ((h+c)^\n) c= dom f
  by A19,A7,A2;
  R is total by Def5;
  then dom R = the carrier of S by PARTFUN1:def 2;
  then
A27: rng (h^\n) c= dom R;
A28: rng (c^\n) c= dom f
  proof
    let y be object;
    assume
A29: y in rng (c^\n);
    rng (c^\n) = rng c by VALUED_0:26;
    then y = x0 by A4,A29,TARSKI:def 1;
    then y in N by NFCONT_1:4;
    hence thesis by A2;
  end;
A30: dom modetrans(L,S,T) = the carrier of S by FUNCT_2:def 1;
  then
A31: rng (h^\n) c= dom modetrans(L,S,T);
  now
    let k;
    thus (f/*((h+c)^\n)-f/*(c^\n)).k = (f/*((h+c)^\n)).k-(f/*(c^\n)).k by
NORMSP_1:def 3
      .= f/.(((h+c)^\n).k) - (f/*(c^\n)).k by A26,FUNCT_2:109
      .= f/.(((h+c)^\n).k) - f/.((c^\n).k) by A28,FUNCT_2:109
      .= L.((h^\n).k) + R/.((h^\n).k) by A21
      .= modetrans(L,S,T).((h^\n).k) + R/.((h^\n).k) by LOPBAN_1:def 11
      .= LP/.((h^\n).k) + R/.((h^\n).k) by A30,PARTFUN1:def 6
      .= (LP/*(h^\n)).k + R/.((h^\n).k) by A31,FUNCT_2:109
      .= (LP/*(h^\n)).k + (R/*(h^\n)).k by A27,FUNCT_2:109
      .= (LP/*(h^\n) + R/*(h^\n)).k by NORMSP_1:def 2;
  end;
  then
A32: f/*((h+c)^\n) - f/*(c^\n) = LP/*(h^\n) + R/*(h^\n) by FUNCT_2:63;
A33: dom modetrans(L,S,T) = the carrier of S by FUNCT_2:def 1;
  LP is_Lipschitzian_on the carrier of S
  proof
    thus the carrier of S c=dom LP by FUNCT_2:def 1;
    set LL=modetrans(L,S,T);
    consider K being Real such that
A34: 0 <= K and
A35: for x being VECTOR of S holds ||. LL.x .|| <= K * ||. x .|| by
LOPBAN_1:def 8;
    take K+1;
A36: 0 + K < 1 + K by XREAL_1:8;
    now
      let x1,x2 be Point of S such that
      x1 in the carrier of S and
      x2 in the carrier of S;
A37:  ||.LL.(x1-x2) .|| <= K * ||.x1-x2.|| by A35;
      0 <= ||.x1-x2.|| by NORMSP_1:4;
      then
A38:  K * ||.x1-x2.|| <= (K+1 ) * ||.x1-x2.|| by A36,XREAL_1:64;
      ||.LP/.x1-LP/.x2 .|| = ||.LL.x1-LP/.x2 .|| by A33,PARTFUN1:def 6
        .= ||.LL.x1+-LL.x2 .|| by A33,PARTFUN1:def 6
        .= ||.LL.x1+(-1)*LL.x2 .|| by RLVECT_1:16
        .= ||.LL.x1+LL.((-1)*x2) .|| by LOPBAN_1:def 5
        .= ||.LL.(x1+(-1)*x2) .|| by VECTSP_1:def 20
        .= ||.LL.(x1-x2) .|| by RLVECT_1:16;
      hence ||.LP/.x1-LP/.x2 .|| <= (K+1) * ||.x1-x2.|| by A37,A38,XXREAL_0:2;
    end;
    hence thesis by A34;
  end;
  then
A39: LP is_continuous_on the carrier of S by NFCONT_1:45;
A40: rng c c= dom f
  proof
    let y be object;
    assume y in rng c;
    then y = x0 by A4,TARSKI:def 1;
    then y in N by NFCONT_1:4;
    hence thesis by A2;
  end;
A41: h^\n is convergent & rng (h^\n) c= the carrier of S by Def4;
  then
A42: LP/*(h^\n) is convergent by A39,A20,NFCONT_1:18;
A43: R/*(h^\n) is convergent by A24,Th24;
  then
A44: (LP/*(h^\n) + R/*(h^\n)) is convergent by A42,NORMSP_1:19;
  LP/.(0.S) =modetrans(L,S,T).(0.S) by A33,PARTFUN1:def 6
    .= modetrans(L,S,T).(0*0.S) by RLVECT_1:10
    .=0*modetrans(L,S,T).0.S by LOPBAN_1:def 5
    .=0.T by RLVECT_1:10;
  then lim (LP/*(h^\n)) = 0.T by A39,A41,A20,NFCONT_1:18;
  then lim (LP/*(h^\n) + R/*(h^\n)) = 0.T+0.T by A43,A25,A42,NORMSP_1:25;
  then
A45: lim (LP/*(h^\n) + R/*(h^\n)) = 0.T by RLVECT_1:4;
  rng (h+c) c= dom f
  by A5,A2;
  then f/*((h+c)^\n) - f/*(c^\n) =(f/*(h+c))^\n - f/*(c^\n) by VALUED_0:27
    .=(f/*(h+c))^\n - (f/*c) ^\n by A40,VALUED_0:27
    .= (f/*(h+c) - f/*c) ^\n by Th16;
  hence thesis by A32,A44,A45,LOPBAN_3:10,11;
end;
