reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th16:
for X,f st X c= dom f holds f|X is continuous
 iff for s1 st rng s1 c= X & s1 is convergent & lim s1 in X
       holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
proof
   let X,f such that
A1: X c= dom f;
   thus f|X is continuous implies for s1 st rng s1 c= X & s1 is convergent &
        lim s1 in X holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
   proof
    assume A2: f|X is continuous;
    hereby let s1 such that
A3:   rng s1 c= X and
A4:   s1 is convergent and
A5:   lim s1 in X;
A6:  dom (f|X) = dom f /\ X by RELAT_1:61
        .= X by A1,XBOOLE_1:28; then
A7:  f|X is_continuous_in (lim s1) by A2,A5;
     now let n be Element of NAT;
A8:   s1.n in rng s1 by VALUED_0:28;
      thus ((f|X)/*s1).n = (f|X)/.(s1.n) by A3,A6,FUNCT_2:109
          .= f/.(s1.n) by A3,A6,A8,PARTFUN2:15
          .= (f/*s1).n by A1,A3,FUNCT_2:109,XBOOLE_1:1;
     end; then
A9:  (f|X)/*s1 = f/*s1 by FUNCT_2:63;
     lim s1 in REAL by XREAL_0:def 1;
     then f/.(lim s1) = (f|X)/.(lim s1) by A5,A6,PARTFUN2:15
       .= lim (f/*s1) by A3,A4,A6,A7,A9;
     hence f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
              by A3,A4,A6,A7,A9;
    end;
   end;
   assume
A10:for s1 st rng s1 c= X & s1 is convergent & lim s1 in X
     holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1);
    now
A11: dom (f|X) = dom f /\ X by RELAT_1:61
      .= X by A1,XBOOLE_1:28;
     let x1 such that
A12:  x1 in dom(f|X);
     now let s1 such that
A13:   rng s1 c= dom (f|X) and
A14:   s1 is convergent and
A15:   lim s1 = x1;
      now let n be Element of NAT;
A16:   s1.n in rng s1 by VALUED_0:28;
       thus ((f|X)/*s1).n = (f|X)/.(s1.n) by A13,FUNCT_2:109
          .= f/.(s1.n) by A13,A16,PARTFUN2:15
          .= (f/*s1).n by A1,A11,A13,FUNCT_2:109,XBOOLE_1:1;
      end; then
A17:  (f|X)/*s1 = f/*s1 by FUNCT_2:63;
     lim s1 in REAL by XREAL_0:def 1;
     then (f|X)/.(lim s1) = f/.(lim s1) by A12,A15,PARTFUN2:15
        .= lim ((f|X)/*s1) by A10,A12,A11,A13,A14,A15,A17;
      hence (f|X)/*s1 is convergent & (f|X)/.x1 = lim ((f|X)/*s1)
               by A10,A12,A11,A13,A14,A15,A17;
     end;
     hence f|X is_continuous_in x1 by A12;
    end;
    hence thesis;
end;
