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 Th27:
  f is_continuous_in x0 iff x0 in dom f & for s1 be sequence of S
  st rng s1 c= dom f & s1 is convergent & lim s1=x0 &
   (for n being Nat holds s1.n<>x0)
  holds f/*s1 is convergent & f/.x0=lim(f/*s1)
proof
  thus f is_continuous_in x0 implies x0 in dom f & for s1 be sequence of S st
rng s1 c= dom f & s1 is convergent & lim s1=x0 &
  (for n being Nat holds s1.n<>x0) holds f
  /*s1 is convergent & f/.x0=lim(f/*s1);
  assume that
A1: x0 in dom f and
A2: for s1 st rng s1 c=dom f & s1 is convergent & lim s1=x0 & (for n being Nat
  holds s1.n<>x0) holds f/*s1 is convergent & f/.x0=lim(f/*s1);
  thus x0 in dom f by A1;
  let s2 be sequence of S such that
A3: rng s2 c=dom f and
A4: s2 is convergent & lim s2=x0;
  now
    per cases;
    suppose
      ex n st for m be Element of NAT st n<=m holds s2.m=x0;
      then consider N be Element of NAT such that
A5:   for m be Element of NAT st N<=m holds s2.m=x0;
A6:   for n holds (s2^\N).n=x0
      proof
        let n;
        s2.(n+N)=x0 by A5,NAT_1:12;
        hence thesis by NAT_1:def 3;
      end;
A7:   rng (s2^\N) c= rng s2 by VALUED_0:21;
A8:   now
        let p be Real such that
A9:     p>0;
         reconsider n=0 as Nat;
        take n;
        let m be Nat such that
        n<=m;
A10:   m in NAT by ORDINAL1:def 12;
        ||.(f/*(s2^\N)).m-f/.x0.|| =||.f/.((s2^\N).m)-f/.x0.|| by A3,A7,
FUNCT_2:109,XBOOLE_1:1,A10
          .=||.f/.x0-f/.x0.|| by A6,A10
          .=||.0.T.|| by RLVECT_1:15
          .=0 by NORMSP_1:1;
        hence ||.(f/*(s2^\N)).m-f/.x0.||< p by A9;
      end;
      then
A11:  f/*(s2^\N) is convergent;
A12:  f/*(s2^\N)=(f/*s2)^\N by A3,VALUED_0:27;
      then
A13:  f/*s2 is convergent by A11,LOPBAN_3:10;
      f/.x0=lim((f/*s2)^\N) by A8,A11,A12,NORMSP_1:def 7;
      hence thesis by A13,LOPBAN_3:9;
    end;
    suppose
A14:  for n ex m be Element of NAT st n<=m & s2.m<>x0;
      defpred P[Nat,set,set] means for n,m be Element of NAT st $2=
      n & $3=m holds n<m & s2.m<>x0 & for k st n<k & s2.k<>x0 holds m<=k;
      defpred P[Nat] means s2.$1<>x0;
      ex m1 be Element of NAT st 0<=m1 & s2.m1<>x0 by A14;
      then
A15:  ex m be Nat st P[m];
      consider M be Nat such that
A16:  P[M] & for n be Nat st P[n] holds M<=n from NAT_1:sch 5(A15);
      reconsider M9 = M as Element of NAT by ORDINAL1:def 12;
A17:  now
        let n;
        consider m be Element of NAT such that
A18:    n+1<=m & s2.m<>x0 by A14;
        take m;
        thus n<m & s2.m<>x0 by A18,NAT_1:13;
      end;
A19:  for n being Nat
       for x be Element of NAT ex y be Element of NAT st P[n,x,y]
      proof
        let n be Nat;
        let x be Element of NAT;
        defpred P[Nat] means x<$1 & s2.$1<>x0;
        ex m be Element of NAT st P[m] by A17;
        then
A20:    ex m be Nat st P[m];
        consider l be Nat such that
A21:    P[l] & for k be Nat st P[k] holds l<=k from NAT_1:sch 5(A20);
        reconsider l as Element of NAT by ORDINAL1:def 12;
        take l;
        thus thesis by A21;
      end;
      consider F be sequence of NAT such that
A22:  F.0=M9 & for n be Nat holds P[n,F.n,F.(n+1)] from
      RECDEF_1:sch 2(A19);
A23:  rng F c= REAL by NUMBERS:19;
A24:  rng F c= NAT;
A25:  dom F=NAT by FUNCT_2:def 1;
      then reconsider F as Real_Sequence by A23,RELSET_1:4;
A26:  now
        let n;
        F.n in rng F by A25,FUNCT_1:def 3;
        hence F.n is Element of NAT by A24;
      end;
      now
        let n be Nat;
A27:      n in NAT by ORDINAL1:def 12;
        F.n is Element of NAT & F.(n+1) is Element of NAT by A26,A27;
        hence F.n<F.(n+1) by A22;
      end;
      then reconsider F as increasing sequence of NAT by SEQM_3:def 6;
A28:  s2*F is convergent & lim (s2*F)=x0 by A4,LOPBAN_3:7,8;
A29:  for n st s2.n<>x0 ex m be Element of NAT st F.m=n
      proof
        defpred P[Nat] means s2.$1<>x0 & for m be Element of NAT holds F.m<>$1;
        assume ex n st P[n];
        then
A30:    ex n be Nat st P[n];
        consider M1 be Nat such that
A31:    P[M1] & for n be Nat st P[n] holds M1<=n from NAT_1:sch 5(A30
        );
        defpred P[Nat] means $1<M1 & s2.$1<>x0 & ex m be Element of NAT st F.m
        =$1;
A32:    ex n be Nat st P[n]
        proof
          take M;
          M<=M1 & M <> M1 by A16,A22,A31;
          hence M<M1 by XXREAL_0:1;
          thus s2.M<>x0 by A16;
          take 0;
          thus thesis by A22;
        end;
A33:    for n be Nat st P[n] holds n<=M1;
        consider MX be Nat such that
A34:    P[MX] & for n be Nat st P[n] holds n<=MX from NAT_1:sch 6(A33
        ,A32);
A35:    for k st MX<k & k<M1 holds s2.k=x0
        proof
          given k such that
A36:      MX<k and
A37:      k<M1 & s2.k<>x0;
          now
            per cases;
            suppose
              ex m be Element of NAT st F.m=k;
              hence contradiction by A34,A36,A37;
            end;
            suppose
              for m be Element of NAT holds F.m<>k;
              hence contradiction by A31,A37;
            end;
          end;
          hence contradiction;
        end;
        consider m be Element of NAT such that
A38:    F.m=MX by A34;
A39:    MX<F.(m+1) & s2.(F.(m+1))<>x0 by A22,A38;
        M1 in NAT by ORDINAL1:def 12;
        then
A40:    F.(m+1)<=M1 by A22,A31,A34,A38;
        now
          assume F.(m+1)<>M1;
          then F.(m+1)<M1 by A40,XXREAL_0:1;
          hence contradiction by A35,A39;
        end;
        hence contradiction by A31;
      end;
A41:  for n being Nat holds (s2*F).n<>x0
      proof
        defpred P[Nat] means (s2*F).$1<>x0;
A42:    for k being Nat st P[k] holds P[k+1]
        proof
          let k be Nat such that
          (s2*F).k<>x0;
          P[k,F.k,F.(k+1)] by A22;
          then s2.(F.(k+1))<>x0;
          hence thesis by FUNCT_2:15;
        end;
A43:    P[0] by A16,A22,FUNCT_2:15;
        thus for n being Nat holds P[n] from NAT_1:sch 2(A43,A42);
      end;
A44:  rng (s2*F) c= rng s2 by VALUED_0:21;
      then rng (s2*F) c= dom f by A3;
      then
A45:  f/*(s2*F) is convergent & f/.x0=lim(f/*(s2*F)) by A2,A41,A28;
A46:  now
        let p be Real;
        assume
A47:    0<p;
        then consider n being Nat such that
A48:    for m be Nat st n<=m holds ||.(f/*(s2*F)).m-f/.x0
        .||<p by A45,NORMSP_1:def 7;
         reconsider k=F.n as Nat;
        take k;
        let m be Nat such that
A49:    k<=m;
A50:   m in NAT by ORDINAL1:def 12;
        now
          per cases;
          suppose
A51:        s2.m=x0;
            ||.(f/*s2).m-f/.x0.|| =||.f/.(s2.m)-f/.x0.|| by A3,FUNCT_2:109,A50
              .=||.0.T.|| by A51,RLVECT_1:15
              .=0 by NORMSP_1:1;
            hence ||.(f/*s2).m-f/.x0.||<p by A47;
          end;
          suppose
            s2.m<>x0;
            then consider l be Element of NAT such that
A52:        m=F.l by A29,A50;
            n<=l by A49,A52,SEQM_3:1;
            then ||.(f/*(s2*F)).l-f/.x0.||<p by A48;
            then ||.f/.((s2*F).l)-f/.x0.||<p by A3,A44,FUNCT_2:109,XBOOLE_1:1;
            then ||.f/.(s2.m)-f/.x0.||<p by A52,FUNCT_2:15;
            hence ||.(f/*s2).m-f/.x0.||<p by A3,FUNCT_2:109,A50;
          end;
        end;
        hence ||.(f/*s2).m-f/.x0.||<p;
      end;
      hence f/*s2 is convergent;
      hence f/.x0=lim(f/*s2) by A46,NORMSP_1:def 7;
    end;
  end;
  hence thesis;
end;
