reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;
reserve x for Point of T;

theorem Th53:
  x is_a_condensation_point_of A \/ B implies x
  is_a_condensation_point_of A or x is_a_condensation_point_of B
proof
  assume
A1: x is_a_condensation_point_of A \/ B;
  assume that
A2: not x is_a_condensation_point_of A and
A3: not x is_a_condensation_point_of B;
  consider N1 being a_neighborhood of x such that
A4: N1 /\ A is countable by A2;
  consider N2 being a_neighborhood of x such that
A5: N2 /\ B is countable by A3;
  reconsider N3 = N1 /\ N2 as a_neighborhood of x by CONNSP_2:2;
  N3 /\ A c= N1 /\ A & N3 /\ B c= N2 /\ B by XBOOLE_1:17,26;
  then (N3 /\ A) \/ (N3 /\ B) is countable by A4,A5,CARD_2:85;
  then N3 /\ (A \/ B) is countable by XBOOLE_1:23;
  hence thesis by A1;
end;
