
theorem Th13:
  for X be set for Y be empty Subset-Family of X holds Components( Y) = {X}
proof
  let X be set;
  let Y be empty Subset-Family of X;
  consider p be FinSequence of bool X such that
A1: len p = card Y and
A2: rng p = Y and
A3: Components(Y) = { Intersect (rng MergeSequence(p,q)) where q is
  FinSequence of BOOLEAN : len q = len p } by Def2;
  thus Components(Y) = {X}
  proof
    thus Components(Y) c= {X}
    proof
      let z be object;
      assume z in Components(Y);
      then consider q be FinSequence of BOOLEAN such that
A4:   z = Intersect (rng MergeSequence(p,q)) and
      len q = len p by A3;
      p = <*>(bool X) by A2;
      then rng MergeSequence(p,q) = {} by Th5,RELAT_1:38;
      then Intersect (rng MergeSequence(p,q)) = X by SETFAM_1:def 9;
      hence thesis by A4,TARSKI:def 1;
    end;
    let z be object;
    p = <*>(bool X) by A2;
    then rng MergeSequence(p,<*>(BOOLEAN)) = {} by Th5,RELAT_1:38;
    then
A5: Intersect (rng MergeSequence(p,<*>(BOOLEAN))) = X by SETFAM_1:def 9;
    assume z in {X};
    then z = X by TARSKI:def 1;
    hence thesis by A1,A3,A5;
  end;
end;
