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;

theorem Th43:
  for x being Point of X holds f is_continuous_at x iff 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 x be Point of X;
  thus f is_continuous_at x implies 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
    assume
A1: f is_continuous_at x;
    let G be Subset of Y;
    assume G is open & f.x in G;
    then reconsider G0 = G as a_neighborhood of f.x by CONNSP_2:3;
    consider H0 being a_neighborhood of x such that
A2: f.:H0 c= G0 by A1;
    consider H being Subset of X such that
A3: H is open and
A4: H c= H0 and
A5: x in H by CONNSP_2:6;
    take H;
    f.:H c= f.:H0 by A4,RELAT_1:123;
    hence thesis by A2,A3,A5,XBOOLE_1:1;
  end;
  assume
A6: 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;
  let G0 be a_neighborhood of f.x;
  consider G being Subset of Y such that
A7: G is open and
A8: G c= G0 and
A9: f.x in G by CONNSP_2:6;
  consider H being Subset of X such that
A10: H is open & x in H and
A11: f.:H c= G by A6,A7,A9;
  reconsider H0 = H as a_neighborhood of x by A10,CONNSP_2:3;
  take H0;
  thus thesis by A8,A11,XBOOLE_1:1;
end;
