
theorem Th40:
  for V being RealUnitarySpace, A being Subset-Family of V st A c=
  Family_open_set(V) holds union A in Family_open_set(V)
proof
  let V be RealUnitarySpace;
  let A be Subset-Family of V;
  assume
A1: A c= Family_open_set(V);
  for x being Point of V st x in union A
   ex r being Real st r>0 & Ball(x,r) c= union A
  proof
    let x be Point of V;
    assume x in union A;
    then consider W being set such that
A2: x in W and
A3: W in A by TARSKI:def 4;
    reconsider W as Subset of V by A3;
    consider r being Real such that
A4: r>0 and
A5: Ball(x,r) c= W by A1,A2,A3,Def5;
    take r;
    thus r > 0 by A4;
    W c= union A by A3,ZFMISC_1:74;
    hence thesis by A5;
  end;
  hence thesis by Def5;
end;
