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 Th59:
  for A being Subset of X, x being Point of X, x0 being Point of
  X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds f
  is_continuous_at x iff f|X0 is_continuous_at x0
proof
  let A be Subset of X, x be Point of X, x0 be Point of X0 such that
A1: A c= the carrier of X0 and
A2: A is a_neighborhood of x and
A3: x = x0;
  thus f is_continuous_at x implies f|X0 is_continuous_at x0 by A3,Th58;
  thus f|X0 is_continuous_at x0 implies f is_continuous_at x
  proof
    assume
A4: f|X0 is_continuous_at x0;
    for G being Subset of Y st G is open & f.x in G ex H being Subset of X
    st H is open & x in H & f.:H c= G
    proof
      let G be Subset of Y such that
A5:   G is open and
A6:   f.x in G;
      (f|X0).x0 in G by A3,A6,FUNCT_1:49;
      then consider H0 being Subset of X0 such that
A7:   H0 is open and
A8:   x0 in H0 and
A9:   (f|X0).:H0 c= G by A4,A5,Th43;
      consider V being Subset of X such that
A10:  V is open and
A11:  V c= A and
A12:  x in V by A2,CONNSP_2:6;
      reconsider V0 = V as Subset of X0 by A1,A11,XBOOLE_1:1;
A13:  H0 /\ V0 c= V by XBOOLE_1:17;
      then reconsider H = H0 /\ V0 as Subset of X by XBOOLE_1:1;
A14:  for z being Point of Y holds z in f.:H implies z in G
      proof
        set g = f|X0;
        let z be Point of Y;
        assume z in f.:H;
        then consider y being Point of X such that
A15:    y in H and
A16:    z = f.y by FUNCT_2:65;
        y in V by A13,A15;
        then y in A by A11;
        then
A17:    z = g.y by A1,A16,FUNCT_1:49;
        H0 /\ V0 c= H0 by XBOOLE_1:17;
        then z in g.:H0 by A15,A17,FUNCT_2:35;
        hence thesis by A9;
      end;
      take H;
      V0 is open by A10,TOPS_2:25;
      then H0 /\ V0 is open by A7;
      hence thesis by A3,A8,A10,A12,A13,A14,SUBSET_1:2,TSEP_1:9,XBOOLE_0:def 4;
    end;
    hence thesis by Th43;
  end;
end;
