reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem
  for GX being TopSpace, V being Subset of GX st (not ex A being Subset
  of GX st A is connected & V c= A) holds Component_of V = {}
proof
  let GX be TopSpace, V be Subset of GX such that
A1: not ex A being Subset of GX st A is connected & V c= A;
  consider F being Subset-Family of GX such that
A2: for A being Subset of GX holds A in F iff A is connected & V c= A and
A3: Component_of V = union F by Def1;
  now
    assume F <> {};
    then consider A being Subset of GX such that
A4: A in F by SUBSET_1:4;
    reconsider A as Subset of GX;
    A is connected & V c= A by A2,A4;
    hence contradiction by A1;
  end;
  hence thesis by A3,ZFMISC_1:2;
end;
