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

theorem
  for A being Subset of X st A is open closed & x in A holds A c=
  qComponent_of x implies A = qComponent_of x
proof
  let A be Subset of X;
  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;
  assume A is open closed & x in A;
  then A in F by A1;
  then
A3: qComponent_of x c= A by A2,SETFAM_1:3;
  assume A c= qComponent_of x;
  hence thesis by A3,XBOOLE_0:def 10;
end;
