reserve X, Y for RealNormSpace;

theorem Th10:
  for V be Subset of TopSpaceNorm X, V1 be Subset of
  LinearTopSpaceNorm X st V=V1 holds Cl(V)=Cl(V1)
proof
  let V be Subset of TopSpaceNorm X, V1 be Subset of LinearTopSpaceNorm X such
  that
A1: V=V1;
  thus Cl(V) c= Cl(V1)
  proof
    let z be object;
    assume
A2: z in Cl(V);
A3: for D1 be Subset of LinearTopSpaceNorm X st D1 is closed holds V1 c=
    D1 implies z in D1
    proof
      let D1 be Subset of LinearTopSpaceNorm X such that
A4:   D1 is closed;
      reconsider D0=D1 as Subset of X by NORMSP_2:def 4;
      reconsider D2=D1 as Subset of TopSpaceNorm X by NORMSP_2:def 4;
      D0 is closed by A4,NORMSP_2:32;
      then
A5:   D2 is closed by NORMSP_2:15;
      assume V1 c= D1;
      hence thesis by A1,A2,A5,PRE_TOPC:15;
    end;
    z in the carrier of X by A2;
    then z in the carrier of LinearTopSpaceNorm X by NORMSP_2:def 4;
    hence thesis by A3,PRE_TOPC:15;
  end;
  let z be object;
  assume
A6: z in Cl(V1);
A7: for D1 be Subset of TopSpaceNorm X st D1 is closed holds V c= D1
  implies z in D1
  proof
    let D1 be Subset of TopSpaceNorm X such that
A8: D1 is closed;
    reconsider D0=D1 as Subset of X;
    reconsider D2=D1 as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    D0 is closed by A8,NORMSP_2:15;
    then
A9: D2 is closed by NORMSP_2:32;
    assume V c= D1;
    hence thesis by A1,A6,A9,PRE_TOPC:15;
  end;
  z in the carrier of LinearTopSpaceNorm X by A6;
  then z in the carrier of TopSpaceNorm X by NORMSP_2:def 4;
  hence thesis by A7,PRE_TOPC:15;
end;
