
theorem Th4:
  for GX being non empty TopSpace, A being Subset of GX, B being
  non empty Subset of GX holds A is_a_component_of B implies A <> {}
proof
  let GX be non empty TopSpace, A be Subset of GX, B be non empty Subset of GX;
  assume A is_a_component_of B;
  then consider B1 being Subset of GX|B such that
A1: B1 = A and
A2: B1 is a_component by CONNSP_1:def 6;
  B1 <> {}(GX|B) by A2,CONNSP_1:32;
  hence thesis by A1;
end;
