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 A,B being Subset of GX st A is connected & B is connected & A
  meets B holds A \/ B c= Component_of A & A \/ B c= Component_of B & A c=
  Component_of B & B c= Component_of A
proof
  let A,B be Subset of GX;
A1: A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
A2: for C,D being Subset of GX st C is connected & D is connected & C /\ D
  <>{} holds C \/ D c= Component_of C
  proof
    let C,D be Subset of GX;
    assume that
A3: C is connected and
A4: D is connected and
A5: C /\ D <>{};
    C meets D by A5;
    then
A6: C \/ D is connected by A3,A4,CONNSP_1:1,17;
    C <>{} by A5;
    hence thesis by A3,A6,Th14;
  end;
  assume A is connected & B is connected & A /\ B <>{};
  then A \/ B c= Component_of A & A \/ B c= Component_of B by A2;
  hence thesis by A1;
end;
