reserve p for Real;
reserve S,T for RealNormSpace;
reserve x0 for Point of S;
reserve f for PartFunc of S,T;
reserve c for constant sequence of S;
reserve R for RestFunc of S,T;

theorem Th3:
  for x0 be Point of S for N be Neighbourhood of x0 st N c= dom f
  for z be Point of S for dv be Point of T holds
  ( (for h be 0-convergent non-zero
Real_Sequence for c st ( rng c = {x0} & rng (h*z+c) c= N ) holds h"(#)(f/*(h*z+
  c) - f/*c) is convergent &
dv = lim (h"(#)(f/*(h*z+c) - f/*c))) iff for e be
Real st e > 0
holds ex d be Real st d > 0 & for h be Real st |.h.|
< d & h <>
  0 & h*z+x0 in N holds ||. h"*(f/.(h*z+x0) - f/.x0) - dv .|| < e )
proof
  let x0 be Point of S;
  let N be Neighbourhood of x0 such that
A1: N c= dom f;
  let z be Point of S;
  let dv be Point of T;
A2: now
    reconsider c = NAT --> x0 as sequence of S;
    assume
A3: for h be 0-convergent non-zero Real_Sequence for c st rng c = {x0} & rng
(h*z+c) c= N holds h"(#)(f/*(h*z+c) - f/*c) is convergent & dv = lim (h"(#)(f/*
    (h*z+c) - f/*c));
    now
      let x be object;
      assume x in {x0};
      then x=x0 by TARSKI:def 1;
      then x=c.1;
      hence x in rng c by NFCONT_1:6;
    end;
    then
A4: {x0} c= rng c;
    now
      let x be object;
      assume x in rng c;
      then consider n be Nat such that
A5:     x=c.n by NFCONT_1:6;
      n in NAT by ORDINAL1:def 12;
      then x=x0 by FUNCOP_1:7,A5;
      hence x in {x0} by TARSKI:def 1;
    end;
    then rng c c= {x0};
    then
A6: rng c = {x0} by A4,XBOOLE_0:def 10;
    assume not ( for r be Real st
    r > 0 holds ex d be Real st d > 0 & for h
be Real st |.h.| < d & h <> 0 & h*z+x0 in N
   holds ||. h"*(f/.(h*z+x0) - f/.x0)
    - dv .|| < r );
    then consider r be Real such that
A7: r > 0 and
A8: for d be Real st d > 0
    holds ex h be Real st |.h.| < d & h <> 0 &
    h*z+x0 in N & not ||. h"*(f/.(h*z+x0) - f/.x0) - dv .|| < r;
    defpred P[Nat,Real] means
    ex rr being Real st rr = $2 & |.rr.| < (
1/($1+1)) & rr <> 0 & rr*z+x0 in N & not ( ||. rr"*(f/.(rr*z+x0) - f/.x0) - dv
    .|| < r );
A9: for n be Element of NAT ex h be Element of REAL st P[n,h]
    proof
      let n be Element of NAT;
      0 < 1 * (n + 1)";
      then 0 < 1/(n + 1) by XCMPLX_0:def 9;
      then consider h be Real such that
A10:   |.h.| < 1/(n + 1) & h <> 0 & h*z+x0 in N & not ||.
      h"*(f/.(h*z+x0) - f/.x0) - dv .|| < r by A8;
      h in REAL by XREAL_0:def 1;
      hence thesis by A10;
    end;
    consider h be Real_Sequence such that
A11: for n be Element of NAT holds P[n,h.n] from FUNCT_2:sch 3(A9);
A12: now
      let p be Real;
      assume
A13:  0<p;
      consider n being Nat such that
A14:  p"<n by SEQ_4:3;
      p" + 0 < n + 1 by A14,XREAL_1:8;
      then 1/(n+1) < 1/p" by A13,XREAL_1:76;
      then
A15:  1/(n+1) < p by XCMPLX_1:216;
      take n;
      let m be Nat;
      assume n<=m;
      then
A16:  n + 1 <= m + 1 by XREAL_1:6;
      m in NAT by ORDINAL1:def 12;
      then
A17:  P[m,h.m] by A11;
      1/(m+1) <= 1/(n+1) by A16,XREAL_1:118;
      then |.h.m-0 .| < 1/(n+1) by A17,XXREAL_0:2;
      hence |.h.m-0 .| <p by A15,XXREAL_0:2;
    end;
    then
A18: h is convergent by SEQ_2:def 6;
    then
A19: lim h = 0 by A12,SEQ_2:def 7;
    for n be Nat holds h.n <> 0
    proof
      let n be Nat;
      n in NAT by ORDINAL1:def 12;
      then P[n,h.n] by A11;
      hence thesis;
    end;
    then h is non-zero by SEQ_1:5;
    then reconsider h as 0-convergent non-zero Real_Sequence
    by A18,A19,FDIFF_1:def 1;
    now
      let x be object;
      assume x in rng (h*z+c);
      then consider n be Nat such that
A20:  x=(h*z+c).n by NFCONT_1:6;
A21:   n in NAT by ORDINAL1:def 12;
A22:  x=(h*z).n+c.n by A20,NORMSP_1:def 2
        .=(h.n)*z+c.n by NDIFF_1:def 3
        .=(h.n)*z+x0 by FUNCOP_1:7,A21;
      P[n,h.n] by A11,A21;
      hence x in N by A22;
    end;
    then
A23: rng (h*z+c) c= N;
    then
    h"(#)(f/*(h*z+c) - f/*c) is convergent & dv = lim (h"(#)(f/*(h*z+c) -
    f/*c)) by A3,A6;
    then consider n be Nat such that
A24: for m be Nat st n <=m holds ||. (h"(#)(f/*(h*z+c) - f
    /*c)).m- dv.|| < r by A7,NORMSP_1:def 7;
    x0 in N by NFCONT_1:4;
    then
A25: rng c c= dom f by A1,A6,ZFMISC_1:31;
A26:  n in NAT by ORDINAL1:def 12;
A27: P[n,h.n] by A11,A26;
    ||. (h"(#)(f/*(h*z+c) - f/*c)).n- dv.|| = ||.(h".n)*((f/*(h*z+c) - f
    /*c).n)- dv.|| by NDIFF_1:def 2
      .= ||.(h.n)"*((f/*(h*z+c) - f/*c).n)- dv.|| by VALUED_1:10
      .= ||.(h.n)"*( (f/*(h*z+c)).n - (f/*c).n)- dv.|| by NORMSP_1:def 3
      .= ||.(h.n)"*( (f/*(h*z+c)).n - f/.(c.n)) - dv.|| by A25,FUNCT_2:109,A26
      .= ||.(h.n)"*( (f/*(h*z+c)).n - f/.x0) - dv.|| by FUNCOP_1:7,A26
      .= ||.(h.n)"*( f/.( (h*z+c).n) - f/.x0) - dv.|| by A1,A23,FUNCT_2:109,A26
,XBOOLE_1:1
      .= ||.(h.n)"*( f/.((h*z).n+c.n) - f/.x0) - dv.|| by NORMSP_1:def 2
      .= ||.(h.n)"*( f/.((h*z).n+x0) - f/.x0) - dv.|| by FUNCOP_1:7,A26
      .= ||.(h.n)"*(f/.((h.n)*z+x0) - f/.x0)-dv .|| by NDIFF_1:def 3;
    hence
    for e be Real
   st e > 0 holds ex d be Real st d > 0 & for h be
Real st
|.h.| < d & h <> 0 & h*z+x0 in N holds ||. h"*(f/.(h*z+x0) - f/.x0) - dv .|| <
    e by A24,A27;
  end;
  now
    assume
A28: for e be Real st e > 0 holds
    ex d be Real st d > 0 & for h be Real st
    |.h.| < d & h <> 0 & h*z+x0 in N holds ||. h"*(f/.(h*z+x0) - f/.x0) -
    dv .|| < e;
    now
      let h be 0-convergent non-zero Real_Sequence;
      let c such that
A29:  rng c = {x0} and
A30:  rng (h*z+c) c= N;
A31:  h is convergent & lim h = 0;
      x0 in N by NFCONT_1:4;
      then
A32:  rng c c= dom f by A1,A29,ZFMISC_1:31;
A33:  for n be Element of NAT holds c.n=x0
      proof
        let n be Element of NAT;
        c.n in rng c by NFCONT_1:6;
        hence thesis by A29,TARSKI:def 1;
      end;
A34:  now
        let r be Real;
        assume r > 0;
        then consider d be Real such that
A35:    d > 0 and
A36:    for h be Real st
        |.h.| < d & h <> 0 & h*z+x0 in N holds ||.
        h"*(f/.(h*z+x0) - f/.x0) - dv .|| < r by A28;
        consider n be Nat such that
A37:    for m be Nat st n <=m holds |.h.m-0 .| < d by A31,A35,
SEQ_2:def 7;
        take n;
        thus for m be Nat st n <=m holds ||. (h"(#)(f/*(h*z+c) - f
        /*c)).m- dv.|| < r
        proof
          let m be Nat;
A38:        m in NAT by ORDINAL1:def 12;
A39:      h.m <> 0 by SEQ_1:5;
          assume n <= m;
          then
A40:      |.h.m-0 .| < d by A37;
          (h*z+c).m in rng (h*z+c) by NFCONT_1:6;
          then (h*z+c).m in N by A30;
          then (h*z).m+c.m in N by NORMSP_1:def 2;
          then h.m*z + c.m in N by NDIFF_1:def 3;
          then
A41:      h.m*z + x0 in N by A33,A38;
          ||. (h"(#)(f/*(h*z+c) - f/*c)).m- dv.|| = ||.(h".m)*((f/*(h*z+c
          ) - f/*c).m)- dv.|| by NDIFF_1:def 2
            .= ||.(h.m)"*((f/*(h*z+c) - f/*c).m)- dv.|| by VALUED_1:10
            .= ||.(h.m)"*( (f/*(h*z+c)).m - (f/*c).m)- dv.|| by NORMSP_1:def 3
            .= ||.(h.m)"*( (f/*(h*z+c)).m - f/.(c.m)) - dv.|| by A32,
FUNCT_2:109,A38
            .= ||.(h.m)"*( (f/*(h*z+c)).m - f/.x0) - dv.|| by A33,A38
            .= ||.(h.m)"*( f/.( (h*z+c).m) - f/.x0) - dv.|| by A1,A30,
FUNCT_2:109,XBOOLE_1:1,A38
            .= ||.(h.m)"*( f/.((h*z).m+c.m) - f/.x0) - dv.|| by NORMSP_1:def 2
            .= ||.(h.m)"*( f/.((h*z).m+x0) - f/.x0) - dv.|| by A33,A38
            .= ||.(h.m)"*(f/.((h.m)*z+x0) - f/.x0)-dv .|| by NDIFF_1:def 3;
          hence thesis by A36,A40,A39,A41;
        end;
      end;
      hence (h"(#)(f/*(h*z+c) - f/*c)) is convergent by NORMSP_1:def 6;
      hence lim (h"(#)(f/*(h*z+c) - f/*c)) = dv by A34,NORMSP_1:def 7;
    end;
    hence
    for h be 0-convergent non-zero Real_Sequence for c st rng c = {x0} & rng (h
*z+c) c= N holds h"(#)(f/*(h*z+c) - f/*c) is convergent & dv = lim (h"(#)(f/*(h
    *z+c) - f/*c));
  end;
  hence thesis by A2;
end;
