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 Th44:
  for x0 be Point of S holds f is_differentiable_in x0 implies f
  is_continuous_in x0
proof
  let x0 be Point of S;
  assume
A1: f is_differentiable_in x0;
  then consider N being Neighbourhood of x0 such that
A2: N c= dom f and
  ex L,R st for x be Point of S st x in N holds f/.x - f/.x0 = L.(x-x0 ) +
  R/.(x-x0);
A3: now
    consider g be Real such that
A4: 0<g and
A5: {y where y is Point of S:||.y-x0.|| < g } c= N by NFCONT_1:def 1;
    reconsider s2 = NAT --> x0 as sequence of S;
    let s1 be sequence of S such that
A6: rng s1 c= dom f and
A7: s1 is convergent and
A8: lim s1 = x0 and
A9: for n being Nat holds s1.n<>x0;
    consider l be Nat such that
A10: for m be Nat st l<=m holds ||.s1.m-x0.||<g by A7,A8,A4,
NORMSP_1:def 7;
    deffunc G(Nat) = s1.$1-s2.$1;
    consider s3 be sequence of S such that
A11: for n holds s3.n=G(n) from FUNCT_2:sch 4;
A12: for n being Nat holds s3.n=G(n)
      proof let n be Nat;
        n in NAT by ORDINAL1:def 12;
       hence thesis by A11;
      end;
A13: now
      given n such that
A14:  s3.n=0.S;
      s1.n-s2.n=0.S by A12,A14;
      then s1.n-x0=0.S;
      hence contradiction by A9,RLVECT_1:21;
    end;
    now
      given n being Nat such that
A15:  (s3^\l).n=0.S;
A16:   n+l in NAT by ORDINAL1:def 12;
      s3.(n+l)=0.S by A15,NAT_1:def 3;
      hence contradiction by A13,A16;
    end;
    then
A17: s3^\l is non-zero by Th7;
    reconsider c =s2^\l as constant sequence of S;
A18: s3=s1-s2 by A12,NORMSP_1:def 3;
A19: s2 is convergent by Th18;
    then
A20: s3 is convergent by A7,A18,NORMSP_1:20;
    then
A21: s3^\l is convergent by LOPBAN_3:9;
    lim s2 = s2.0 by Th18
      .=x0;
    then lim s3 =x0-x0 by A7,A8,A19,A18,NORMSP_1:26
      .=0.S by RLVECT_1:15;
    then lim(s3^\l)=0.S by A20,LOPBAN_3:9;
    then reconsider h=s3^\l as (0.S)-convergent sequence of S
      by A21,Def4;

    now
      let n;
      thus (f/*(h+c)-f/*c+f/*c).n =(f/*(h+c)-f/*c).n+(f/*c).n by NORMSP_1:def 2
        .=(f/*(h+c)).n-(f/*c).n+(f/*c).n by NORMSP_1:def 3
        .=(f/*(h+c)).n-((f/*c).n-(f/*c).n) by RLVECT_1:29
        .=(f/*(h+c)).n-0.T by RLVECT_1:15
        .=(f/*(h+c)).n by RLVECT_1:13;
    end;
    then
A22: f/*(h+c)-f/*c+(f/*c)=f/*(h+c) by FUNCT_2:63;
    now
      let n;
      thus (h+c).n=((s1-s2+s2)^\l).n by A18,Th15
        .=(s1-s2+s2).(n+l) by NAT_1:def 3
        .=(s1-s2).(n+l)+s2.(n+l) by NORMSP_1:def 2
        .=s1.(n+l)-s2.(n+l)+s2.(n+l) by NORMSP_1:def 3
        .=s1.(n+l)-(s2.(n+l)-s2.(n+l)) by RLVECT_1:29
        .=s1.(n+l)-0.S by RLVECT_1:15
        .=s1.(l+n) by RLVECT_1:13
        .=(s1^\l).n by NAT_1:def 3;
    end;
    then
A23: f/*(h+c)-f/*c+(f/*c)=f/*(s1^\l) by A22,FUNCT_2:63
      .=(f/*s1)^\l by A6,VALUED_0:27;
A24: rng c = {x0}
    proof
      thus rng c c= {x0}
      proof
        let y be object;
        assume y in rng c;
        then consider n being Nat such that
A25:    y=(s2^\l).n by NFCONT_1:6;
A26:     n+l in NAT by ORDINAL1:def 12;
        y=s2.(n+l) by A25,NAT_1:def 3;
        then y=x0 by FUNCOP_1:7,A26;
        hence thesis by TARSKI:def 1;
      end;
      let y be object;
      assume y in {x0};
      then
A27:  y=x0 by TARSKI:def 1;
A28:     0+l in NAT by ORDINAL1:def 12;
      c.0=s2.(0+l) by NAT_1:def 3
        .= y by A27,FUNCOP_1:7,A28;
      hence thesis by NFCONT_1:6;
    end;
A29: now
      let p be Real such that
A30:  0<p;
       reconsider n=0 as Nat;
      take n;
      let m be Nat such that
      n<=m;
A31:   m in NAT by ORDINAL1:def 12;
A32:   m+l in NAT by ORDINAL1:def 12;
      x0 in N by NFCONT_1:4;
      then rng c c= dom f by A2,A24,ZFMISC_1:31;
      then ||.(f/*c).m-f/.x0.|| =||.f/.(c.m)-f/.x0.|| by FUNCT_2:109,A31
        .=||.f/.(s2.(m+l))-f/.x0.|| by NAT_1:def 3
        .=||.f/.x0-f/.x0.|| by FUNCOP_1:7,A32
        .=||.0.T.|| by RLVECT_1:15
        .=0 by NORMSP_1:1;
      hence ||.(f/*c).m-f/.x0.||<p by A30;
    end;
    then
A33: f/*c is convergent;
A34: rng (h+c) c= N
    proof
      let y be object;
      assume
A35:  y in rng(h+c);
      then consider n being Nat such that
A36:  y=(h+c).n by NFCONT_1:6;
      reconsider y1=y as Point of S by A35;
      (h+c).n=((s1-s2+s2)^\l).n by A18,Th15
        .=(s1-s2+s2).(n+l) by NAT_1:def 3
        .=(s1-s2).(n+l)+s2.(n+l) by NORMSP_1:def 2
        .=s1.(n+l)-s2.(n+l)+s2.(n+l) by NORMSP_1:def 3
        .=s1.(n+l)-(s2.(n+l)-s2.(n+l)) by RLVECT_1:29
        .=s1.(n+l)-0.S by RLVECT_1:15
        .=s1.(l+n) by RLVECT_1:13;
      then ||.(h+c).n-x0.||<g by A10,NAT_1:12;
      then y1 in {z where z is Point of S:||.z-x0.|| < g } by A36;
      hence thesis by A5;
    end;
    then
A37: (f/*(h+c) - f/*c) is convergent by A17,A1,A2,A24,Th34;
    then f/*(h+c)-f/*c+f/*c is convergent by A33,NORMSP_1:19;
    hence f/*s1 is convergent by A23,LOPBAN_3:10;
A38: lim (f/*(h+c) - f/*c) = 0.T by A17,A1,A2,A24,A34,Th34;
    lim(f/*c)=f/.x0 by A29,A33,NORMSP_1:def 7;
    then lim(f/*(h+c)-f/*c+f/*c)=0.T +f/.x0 by A37,A38,A33,NORMSP_1:25
      .=f/.x0 by RLVECT_1:4;
    hence f/.x0=lim(f/*s1) by A37,A33,A23,LOPBAN_3:11,NORMSP_1:19;
  end;
  x0 in N by NFCONT_1:4;
  hence thesis by A2,A3,Th27;
end;
