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 Th21:
  for A being Subset of GX,p being Point of GX st p in A & A is
  a_union_of_components of GX holds Component_of p c= A
proof
  let A be Subset of GX,p be Point of GX;
  assume that
A1: p in A and
A2: A is a_union_of_components of GX;
  consider F being Subset-Family of GX such that
A3: for B being Subset of GX st B in F holds B is a_component and
A4: A=union F by A2,Def2;
  consider X such that
A5: p in X and
A6: X in F by A1,A4,TARSKI:def 4;
  reconsider B2=X as Subset of GX by A6;
  B2=Component_of p by A3,A5,A6,CONNSP_1:41;
  hence thesis by A4,A6,ZFMISC_1:74;
end;
