 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem  Th33:
for X, X1 being set
for S being RealNormSpace
for f being PartFunc of S,REAL
st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
proof
  let X, X1 be set;
  let S be RealNormSpace;
  let f be PartFunc of S,REAL;
  assume that
A1: f is_continuous_on X and
A2: X1 c= X;
  X c= dom f by A1;
  hence A3: X1 c= dom f by A2;
  let r be Point of S;
  assume A4: r in X1; then
A5: f | X is_continuous_in r by A1, A2;
  thus f | X1 is_continuous_in r
  proof
    (dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
    then dom (f | X1) c= (dom f) /\ X by RELAT_1:61; then
A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
    r in (dom f) /\ X1 by A3, A4, XBOOLE_0:def 4;
    hence A7: r in dom (f | X1) by RELAT_1:61; then
A8: (f | X) /. r = f /. r by A6, PARTFUN2:15
      .= (f | X1) /. r by A7, PARTFUN2:15;
    let s1 be sequence of S;
    assume that
A9: rng s1 c= dom (f | X1) and
A10:s1 is convergent & lim s1 = r;
A11:rng s1 c= dom (f | X) by A9, A6;
A12: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 | X) /* s1) . n
        = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
       .= f /. (s1 . n) by A9, A6,A13, PARTFUN2:15
       .= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
       .= ((f | X1) /* s1) . n by A9, FUNCT_2:109;
    end;
    ( (f | X) /* s1 is convergent
      & (f | X) /. r = lim ((f | X) /* s1) ) by A5, A10, A11;
    hence ( (f | X1) /* s1 is convergent
      & (f | X1) /. r = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63;
  end;
end;
