 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th1:
 for X be RealNormSpace,
     Y be Subset of X,
     Z be Subset of MetricSpaceNorm X
  st Y = Z
holds Cl(Y) = Cl(Z)
proof
  let X be RealNormSpace,
      Y be Subset of X,
      Z be Subset of MetricSpaceNorm X;
  assume A1: Y =Z;
  consider S be Subset of LinearTopSpaceNorm X such that
     A2: Y = S & Cl Y = Cl S by NORMSP_3:def 1;
A3:the carrier of TopSpaceNorm X = the carrier of LinearTopSpaceNorm X
  by NORMSP_2:def 4;
consider D be Subset of TopSpaceMetr MetricSpaceNorm X such that
      A4: D = Z & Cl(Z) = Cl D by Def1;
 for p being set st p in the carrier of LinearTopSpaceNorm X holds
( p in Cl D iff for G being Subset of LinearTopSpaceNorm X
      st G is open & p in G holds S meets G )
proof
  let p be set;
  assume
 A5:  p in the carrier of LinearTopSpaceNorm X;
  ( for G being Subset of TopSpaceMetr MetricSpaceNorm X
      st G is open & p in G holds D meets G )
iff
(for G being Subset of LinearTopSpaceNorm X
      st G is open & p in G holds S meets G)
proof
hereby assume
A6:( for G being Subset of TopSpaceMetr MetricSpaceNorm X
      st G is open & p in G holds D meets G );
  let G be Subset of LinearTopSpaceNorm X;
  assume A7: G is open & p in G;
  reconsider G0 = G as Subset of TopSpaceMetr MetricSpaceNorm X
    by NORMSP_2:def 4;
  G0 is open & p in G0 by A7,A3,NORMSP_2:def 4;
  hence S meets G by A6,A2,A4,A1;
end;
assume
A9: (for G being Subset of LinearTopSpaceNorm X
      st G is open & p in G holds S meets G);
  let G be Subset of TopSpaceMetr MetricSpaceNorm X;
  assume A10: G is open & p in G;
  reconsider G0 = G as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
  G0 is open & p in G0 by A10,A3,NORMSP_2:def 4;
  hence D meets G by A9,A2,A4,A1;
end;
hence thesis by A5,PRE_TOPC:def 7,A3;
end;
hence thesis by A2,A4,PRE_TOPC:def 7,A3;
end;
