reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;

theorem Th58:
  for x being Point of X, x0 being Point of X0 st x = x0 holds f
  is_continuous_at x implies f|X0 is_continuous_at x0
proof
  let x be Point of X, x0 be Point of X0 such that
A1: x = x0;
  assume
A2: f is_continuous_at x;
  for G being Subset of Y st G is open & (f|X0).x0 in G ex H0 being Subset
  of X0 st H0 is open & x0 in H0 & (f|X0).:H0 c= G
  proof
    reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
    let G be Subset of Y such that
A3: G is open and
A4: (f|X0).x0 in G;
    f.x in G by A1,A4,FUNCT_1:49;
    then consider H being Subset of X such that
A5: H is open & x in H and
A6: f.:H c= G by A2,A3,Th43;
    reconsider H0 = H /\ C as Subset of X0 by XBOOLE_1:17;
    f.:H0 c= f.:H /\ f.:C & f.:H /\ f.:C c= f.:H by RELAT_1:121,XBOOLE_1:17;
    then f.:H0 c= f.:H by XBOOLE_1:1;
    then
A7: f.:H0 c= G by A6,XBOOLE_1:1;
    take H0;
    H0 = H /\ [#]X0 & (f|X0).:H0 c= f.:H0 by RELAT_1:128;
    hence thesis by A1,A5,A7,TOPS_2:24,XBOOLE_0:def 4,XBOOLE_1:1;
  end;
  hence thesis by Th43;
end;
