reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th2:
for S be RealNormSpace, f be PartFunc of S,REAL
  holds
    f is_continuous_on Z
 iff
    Z c= dom f
  & for s1 be sequence of S
     st rng s1 c= Z & s1 is convergent & lim s1 in Z
     holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
proof
   let S be RealNormSpace, f be PartFunc of S,REAL;
   thus f is_continuous_on Z implies Z c= dom f & for s1 be sequence of S
          st rng s1 c= Z & s1 is convergent & lim s1 in Z
          holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
   proof
    assume A1: f is_continuous_on Z; then
A2: Z c= dom f by NFCONT_1:def 8;
    now let s1 be sequence of S;
     assume A3: rng s1 c= Z & s1 is convergent & lim s1 in Z; then
A4:  f|Z is_continuous_in (lim s1) by A1,NFCONT_1:def 8;
     dom (f|Z) = dom f /\ Z by PARTFUN2:15; then
A5:  dom (f|Z) = Z by A2,XBOOLE_1:28;
     now let n be Element of NAT;
      dom s1 = NAT by FUNCT_2:def 1; then
A6:   s1.n in rng s1 by FUNCT_1:3;
      thus ((f|Z)/*s1).n = (f|Z)/.(s1.n) by A3,A5,FUNCT_2:109
          .= f/.(s1.n) by A3,A5,A6,PARTFUN2:15
          .= (f/*s1).n by A2,A3,FUNCT_2:109,XBOOLE_1:1;
     end; then
A7:  (f|Z)/*s1 = f/*s1;
     f/.(lim s1) = (f|Z)/.(lim s1) by A3,A5,PARTFUN2:15;
     hence f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
        by A3,A5,A4,A7,NFCONT_1:def 6;
    end;
    hence thesis by A1,NFCONT_1:def 8;
   end;
   assume that
A8: Z c= dom f and
A9: for s1 be sequence of S
      st rng s1 c= Z & s1 is convergent & lim s1 in Z
       holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1);
   dom (f|Z) = dom f /\ Z by PARTFUN2:15; then
A10:dom (f|Z) = Z by A8,XBOOLE_1:28;
   now let x1 be Point of S;
    assume A11: x1 in Z;
    now let s1 be sequence of S such that
A12:  rng s1 c= dom (f|Z) & s1 is convergent & lim s1=x1;
     now let n be Element of NAT;
      dom s1 = NAT by FUNCT_2:def 1; then
A13:  s1.n in rng s1 by FUNCT_1:3;
      thus ((f|Z)/*s1).n = (f|Z)/.(s1.n) by A12,FUNCT_2:109
          .= f/.(s1.n) by A12,A13,PARTFUN2:15
          .= (f/*s1).n by A8,A10,A12,FUNCT_2:109,XBOOLE_1:1;
     end; then
A14: (f|Z)/*s1 = f/*s1;
     (f|Z)/.(lim s1) = f/.(lim s1) by A11,A10,A12,PARTFUN2:15;
     hence (f|Z)/*s1 is convergent & (f|Z)/.x1 = lim ((f|Z)/*s1)
        by A9,A11,A10,A12,A14;
    end;
    hence f|Z is_continuous_in x1 by A11,A10,NFCONT_1:def 6;
   end;
   hence thesis by A8,NFCONT_1:def 8;
end;
