reserve i for Integer,
  a, b, r, s for Real;

theorem
  for S, T being non empty TopSpace, X being Subset of S, Y being Subset
of T, f being continuous Function of S,T, g being Function of S|X,T|Y st g = f|
  X holds g is continuous
proof
  let S, T be non empty TopSpace, X be Subset of S, Y be Subset of T, f be
  continuous Function of S,T, g be Function of S|X,T|Y such that
A1: g = f|X;
  set h = f|X;
A2: the carrier of S|X = X & rng h c= the carrier of T by PRE_TOPC:8;
  dom f = the carrier of S by FUNCT_2:def 1;
  then dom h = X by RELAT_1:62;
  then reconsider h as Function of S|X,T by A2,FUNCT_2:2;
  h is continuous by TOPMETR:7;
  hence thesis by A1,TOPMETR:6;
end;
