reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th13:
  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;
    now
      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;
A8:     s1.n in rng s1 by VALUED_0:28;
        thus ((f|X)/*s1).n = (f|X).(s1.n) by A3,A6,FUNCT_2:108
          .= f.(s1.n) by A3,A6,A8,FUNCT_1:47
          .= (f/*s1).n by A1,A3,FUNCT_2:108,XBOOLE_1:1;
      end;
      then
A9:   (f|X)/*s1 = f/*s1 by FUNCT_2:63;
      f.(lim s1) = (f|X).(lim s1) by A5,A6,FUNCT_1:47
        .= 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;
    hence thesis;
  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;
A16:    s1.n in rng s1 by VALUED_0:28;
        thus ((f|X)/*s1).n = (f|X).(s1.n) by A13,FUNCT_2:108
          .= f.(s1.n) by A13,A16,FUNCT_1:47
          .= (f/*s1).n by A1,A11,A13,FUNCT_2:108,XBOOLE_1:1;
      end;
      then
A17:  (f|X)/*s1 = f/*s1 by FUNCT_2:63;
      (f|X).(lim s1) = f.(lim s1) by A12,A15,FUNCT_1:47
        .= 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;
  end;
  hence thesis;
end;
