reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem Th15:
  C c= meet BDD-Family C
proof
  for Z being set st Z in BDD-Family C holds C c= Z
  proof
    let Z be set;
    assume Z in BDD-Family C;
    then consider k being Nat such that
A1: Z = Cl BDD L~Cage(C,k);
    C c= BDD L~Cage(C,k) & BDD L~Cage(C,k) c= Cl BDD L~Cage(C,k) by Th12,
PRE_TOPC:18;
    hence thesis by A1;
  end;
  hence thesis by SETFAM_1:5;
end;
