reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem
  ex R be RestFunc of F st R/.0=0.F & R is_continuous_in 0
  proof
    ([#] REAL)` = {} REAL & REAL c= REAL & [#]REAL = REAL by XBOOLE_1:37;
    then
    reconsider Z = [#]REAL as open Subset of REAL by Lm1,RCOMP_1:def 5;
    set R = (REAL --> 0.F);
    reconsider f=R as PartFunc of REAL,the carrier of F;
A1: dom R = REAL;
    now
      let h;
      now
        let n be Nat;
        A2: rng h c= dom R;
        A3: n in NAT by ORDINAL1:def 12;
        thus ((h")(#)(R/*h)).n = (h".n)*((R/*h).n) by NDIFF_1:def 2
        .= (h".n)*(R/.(h.n)) by A3,A2,FUNCT_2:108
        .= 0.F by RLVECT_1:10;
      end;
      then (h")(#)(R/*h) is constant
      & ((h")(#)(R/*h)).0 = 0.F by VALUED_0:def 18;
      hence (h")(#)(R/*h) is convergent &
      lim ((h")(#)(R/*h)) = 0.F by NDIFF_1:18;
    end;
    then reconsider R as RestFunc of F by Def1;
    set L = (REAL --> 0.F);
    now let p be Real;
       reconsider pp=p as Element of REAL by XREAL_0:def 1;
      thus L/.p = L/.pp
      .= 0.F
      .= p*(0.F) by RLVECT_1:10;
    end;
    then reconsider L as LinearFunc of F by Def2;
A5: f|Z is constant;
    consider r be Point of F such that
A6: for x being Element of REAL  st x in Z/\dom f holds f.x=r
        by A5,PARTFUN2:57;
A7: now let x;
    assume A8: x in Z/\dom f;
    then x in dom f;
    hence f/.x= f.x by PARTFUN1:def 6
    .= r by A8,A6;
  end;
  A9: now
  let x0;
  assume
A10: x0 in Z;
  set N = the Neighbourhood of x0;
  for x st x in N holds f/.x-f/.x0=L/.(x-x0)+R/.(x-x0)
  proof
A12: x0 in Z/\dom f by A10;
    let x;
A13:   x-x0 in REAL by XREAL_0:def 1;
    then
    A14: R/.(x-x0)=R.(x-x0) by A1,PARTFUN1:def 6
    .= 0.F by FUNCOP_1:7,A13;
    assume x in N;
    then x in Z/\dom f;
    hence f/.x-f/.x0=r-f/.x0 by A7
    .=r - r by A7,A12
    .=0.F by RLVECT_1:15
    .=L.(x-x0) by FUNCOP_1:7,A13
    .=L/.(x-x0) by A1,PARTFUN1:def 6,A13
    .=L/.(x-x0) + R/.(x-x0) by A14,RLVECT_1:4;
  end;
  hence f is_differentiable_in x0;
end;
set x0 = the Element of REAL;

f is_differentiable_in x0 by A9; then
consider N being Neighbourhood of x0 such that
N c= dom f and
A15: ex L,R st for x st x in N holds f/.x - f/.x0 = L/.(x-x0) + R/.(x-x0);
consider L,R such that
A16: for x st x in N holds f/.x - f/.x0 = L/.(x-x0) + R/.(x-x0) by A15;
take R;
consider p be Point of F such that
A17: for r holds L/.r = r*p by Def2;
f/.x0 - f/.x0 = L/.(x0-x0) + R/.(x0-x0) by A16,RCOMP_1:16; then
0.F = L/.(x0-x0) + R/.(x0-x0) by RLVECT_1:15; then
0.F = 0 * p+ R/.0 by A17; then
0.F = 0.F + R/.0 by RLVECT_1:10;
hence
A18: R/.0=0.F by RLVECT_1:4;
A19: now
set s3 = cs;
let h;
A20: s3.1 = 0;
(h")(#)(R/*h) is convergent &
lim ((h")(#)(R/*h)) = 0.F by Def1;
then ||.(h")(#)(R/*h).|| is bounded by LOPBAN_1:20,SEQ_2:13;
then consider M being Real such that M>0 and
A21: for n being Nat holds |.(||.((h")(#)(R/*h)) .||).n .| <M by SEQ_2:3;
A22: now let n;
|.(||.((h")(#)(R/*h)) .||).n .| <M by A21; then
|. ||.((h")(#)(R/*h)).n .|| .| <M by NORMSP_0:def 4;
hence ||.((h")(#)(R/*h)).n .|| < M by ABSVALUE:def 1;
end;
reconsider z0=0 as Real;
A23: now
let n be Element of NAT;
||.((h")(#)(R/*h)).n .||=||.((h").n)*(R/*h).n .|| by NDIFF_1:def 2
.=||.(h.n)"*(R/*h).n .|| by VALUED_1:10; then
A24: ||.(h.n)"*(R/*h).n .||<=M by A22;
|.(h.n).|>=0 by COMPLEX1:46;
then |.(h.n).|*||.(h.n)"*(R/*h).n .|| <=M*|.(h.n).| by A24,XREAL_1:64;
then ||.(h.n)*((h.n)"*(R/*h).n) .|| <=M*|.(h.n).| by NORMSP_1:def 1; then
A25: ||.(h.n)*(h.n)"*(R/*h).n .||<=M*|.(h.n).| by RLVECT_1:def 7;
h.n <>0 by SEQ_1:5;
then ||.1*(R/*h).n.|| <=M*|.(h.n).| by A25,XCMPLX_0:def 7;
then ||.(R/*h).n.|| <=M*|.(h.n).| by RLVECT_1:def 8;
then ||.(R/*h).n .|| <=M*abs(h).n by SEQ_1:12;
hence ||.(R/*h).n .|| <=(M(#)abs(h)).n by SEQ_1:9;
end;
lim h=z0;
then lim abs(h) = |.z0.| by SEQ_4:14
.=z0 by ABSVALUE:2; then
A26: lim (M(#)abs(h)) = M*z0 by SEQ_2:8
.=lim s3 by A20,SEQ_4:25;
reconsider z0=0 as Real;
lim (M(#)abs(h)) = 0 by A26,A20,SEQ_4:25;
hence R/*h is convergent & lim (R/*h)=R/.0 by A18,A23,Th1;
end;
R is total by Def1; then
A27:dom R=REAL by FUNCT_2:def 1;
 now
  let s1;
  assume that rng s1 c= dom R and
  A28: s1 is convergent & lim s1 = 0 and
  A29: for n being Nat holds s1.n <> 0;
  for n being Nat holds s1.n <> 0
   by A29;
  then s1 is non-zero by SEQ_1:5;
  then s1 is 0-convergent non-zero Real_Sequence by A28,FDIFF_1:def 1;
  hence R/*s1 is convergent & R/.In(0,REAL)=lim (R/*s1) by A19;
end;
hence thesis by A27,NFCONT_3:7;
end;
