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

theorem
  x in qComponent_of x
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;
  F <> {} & for A being set holds A in F implies x in A by A1,Th22;
  hence thesis by A2,SETFAM_1:def 1;
end;
