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 Th14:
  for A being Subset of GX,B being Subset of GX st A is connected
  & A \/ B is connected & A<>{} holds A \/ B c= Component_of A
proof
  let A be Subset of GX,B be Subset of GX;
  assume that
A1: A is connected and
A2: A \/ B is connected and
A3: A<>{};
  Component_of (A \/ B) = Component_of A by A1,A2,A3,Th12,XBOOLE_1:7;
  hence thesis by A2,Th1;
end;
