
theorem
  for X be RealNormSpace, A being Subset of X
  ex F being Subset-Family of X
  st (for C being Subset of X holds C in F iff C is closed & A c= C)
   & Cl A = meet F
  proof
    let X be RealNormSpace, A be Subset of X;
    reconsider B = A as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    consider G being Subset-Family of LinearTopSpaceNorm X such that
    A1: (for C being Subset of LinearTopSpaceNorm X
      holds C in G iff C is closed & B c= C)
      & Cl B = meet G by PRE_TOPC:16;
    reconsider F = G as Subset-Family of X by NORMSP_2:def 4;
    for C being Subset of X holds C in F iff C is closed & A c= C
    proof
      let C be Subset of X;
      reconsider D = C as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
      D in G iff D is closed & B c= D by A1;
      hence thesis by NORMSP_2:32;
    end;
    hence thesis by A1,EQCL1;
  end;
