reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  qComponent_of x is closed
proof
  consider F being Subset-Family of X such that
A1: for A being Subset of X holds (A in F iff A is open closed & x in A) and
A2: qComponent_of x = meet F by Def7;
  for A being Subset of X st A in F holds A is closed by A1;
  hence thesis by A2,PRE_TOPC:14;
end;
