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 Th1:
  for GX being TopSpace, V being Subset of GX st (ex A being Subset
  of GX st A is connected & V c= A) holds V c= Component_of V
proof
  let GX be TopSpace, V be Subset of GX;
  given A being Subset of GX such that
A1: 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;
A4: for A being set holds A in F implies V c= A by A2;
  F <> {} by A1,A2;
  then
A5: V c= meet F by A4,SETFAM_1:5;
  meet F c= union F by SETFAM_1:2;
  hence thesis by A3,A5;
end;
