theorem
  for T being TopSpace, A being Subset of T, B being Subset of T st B
  is_a_component_of A holds B is connected
proof
  let T be TopSpace, A be Subset of T, B be Subset of T;
  assume B is_a_component_of A;
  then consider C being Subset of T|A such that
A1: C = B and
A2: C is a_component by CONNSP_1:def 6;
  C is connected by A2,CONNSP_1:def 5;
  hence thesis by A1,CONNSP_1:23;
end;
