reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;
reserve GZ for non empty TopSpace;

theorem
  for A,B being Subset of GZ st A is a_component & B
  is a_component holds A \/ B is a_union_of_components of GZ
proof
  let A,B be Subset of GZ;
  {A,B} c= bool (the carrier of GZ)
  proof
    let x be object;
    assume x in {A,B};
    then x=A or x=B by TARSKI:def 2;
    hence thesis;
  end;
  then reconsider F2={A,B} as Subset-Family of GZ;
  reconsider F=F2 as Subset-Family of GZ;
  assume A is a_component & B is a_component;
  then
A1: for B1 being Subset of GZ st B1 in F holds B1 is a_component by
TARSKI:def 2;
  A \/ B=union F by ZFMISC_1:75;
  hence thesis by A1,CONNSP_3:def 2;
end;
