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 Th5:
  for GX being TopSpace, V being Subset of GX st V is connected & V
  <> {} holds Component_of V is connected
proof
  let GX be TopSpace;
  let V be Subset of GX;
  assume that
A1: V is connected and
A2: V<>{};
  consider F being Subset-Family of GX such that
A3: for A being Subset of GX holds A in F iff A is connected & V c= A and
A4: Component_of V = union F by Def1;
A5: for A being set st A in F holds V c= A by A3;
  F <> {} by A1,A3;
  then V c= meet F by A5,SETFAM_1:5;
  then
A6: meet F<>{}(GX) by A2;
  for A being Subset of GX st A in F holds A is connected by A3;
  hence thesis by A4,A6,CONNSP_1:26;
end;
