reserve X for set,
  Y for non empty set;

theorem Th7:
  for T being non empty TopSpace for A,B,C being Subset of T st A
  c= B & A is_a_component_of C & B is_a_component_of C holds A = B
proof
  let T be non empty TopSpace;
  let A,B,C be Subset of T such that
A1: A c= B and
A2: A is_a_component_of C and
A3: B is_a_component_of C;
  per cases;
  suppose
    C = {};
    then
A4: C = {}T;
    then A = {} by A2,Th6;
    hence thesis by A3,A4,Th6;
  end;
  suppose
    C is non empty;
    then A <> {} by A2,SPRECT_1:4;
    hence thesis by A1,A2,A3,GOBOARD9:1,XBOOLE_1:69;
  end;
end;
