reserve a, r, s for Real;

theorem Th3:
  for T being TopStruct, A being Subset of T, t be Point of T st t
  in A holds Component_of(t,A) c= A
proof
  let T be TopStruct, A be Subset of T, t be Point of T;
  assume
A1: t in A;
  then Down(t,A) = t by CONNSP_3:def 3;
  then
A2: Component_of(t,A) = Component_of Down(t,A) by A1,CONNSP_3:def 7;
  the carrier of (T|A) = A by PRE_TOPC:8;
  hence thesis by A2;
end;
