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 Th42:
  for x being Point of X holds f is_continuous_at x iff for G
  being a_neighborhood of f.x holds f"G is a_neighborhood of x
proof
  let x be Point of X;
  thus f is_continuous_at x implies for G being a_neighborhood of f.x holds f"
  G is a_neighborhood of x
  proof
    assume
A1: f is_continuous_at x;
    let G be a_neighborhood of f.x;
    consider H being a_neighborhood of x such that
A2: f.:H c= G by A1;
    ex V being Subset of X st V is open & V c= f"G & x in V
    proof
      consider V being Subset of X such that
A3:   V is open & V c= H & x in V by CONNSP_2:6;
      take V;
      H c= f"G by A2,FUNCT_2:95;
      hence thesis by A3,XBOOLE_1:1;
    end;
    hence thesis by CONNSP_2:6;
  end;
  assume
A4: for G being a_neighborhood of f.x holds f"G is a_neighborhood of x;
  let G be a_neighborhood of f.x;
  reconsider H = f"G as a_neighborhood of x by A4;
  take H;
  thus thesis by FUNCT_1:75;
end;
