reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem Th11:
  for A being Subset of X, B being Subset of X st A is a_component & A c= B
  holds A is_a_component_of B
proof
  let A be Subset of X,B be Subset of X such that
A1: A is a_component and
A2: A c= B;
A3: A is connected by A1;
  ex A1 being Subset of X|B st A=A1 & A1 is a_component
  proof
    B = [#](X|B) by PRE_TOPC:def 5;
    then reconsider C = A as Subset of X|B by A2;
    take A1=C;
A4: for D being Subset of X|B st D is connected holds A1 c= D implies A1 = D
    proof
      let D be Subset of X|B such that
A5:   D is connected;
      reconsider D1=D as Subset of X by PRE_TOPC:11;
      assume
A6:   A1 c= D;
      D1 is connected by A5,CONNSP_1:23;
      hence thesis by A1,A6,CONNSP_1:def 5;
    end;
    A1 is connected by A3,CONNSP_1:23;
    hence thesis by A4,CONNSP_1:def 5;
  end;
  hence thesis by CONNSP_1:def 6;
end;
