reserve X for RealNormSpace;

theorem Th34:
  for X be RealNormSpace, U be Subset of TopSpaceNorm X, Ut be
  Subset of LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be Point of
  LinearTopSpaceNorm X st U=Ut & x=xt holds U is a_neighborhood of x iff Ut is
  a_neighborhood of xt
proof
  let X be RealNormSpace, U be Subset of TopSpaceNorm X, Ut be Subset of
  LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be Point of
  LinearTopSpaceNorm X;
  assume that
A1: U=Ut and
A2: x=xt;
  hereby
    assume U is a_neighborhood of x;
    then consider V be Subset of TopSpaceNorm X such that
A3: V is open and
A4: V c= U and
A5: x in V by CONNSP_2:6;
    reconsider Vt=V as open Subset of LinearTopSpaceNorm X by A3,Def4,Th20;
    Vt c= Ut by A1,A4;
    hence Ut is a_neighborhood of xt by A2,A5,CONNSP_2:6;
  end;
  assume Ut is a_neighborhood of xt;
  then consider Vt be Subset of LinearTopSpaceNorm X such that
A6: Vt is open and
A7: Vt c= Ut and
A8: xt in Vt by CONNSP_2:6;
  reconsider V=Vt as open Subset of TopSpaceNorm X by A6,Def4,Th20;
  V c= U by A1,A7;
  hence thesis by A2,A8,CONNSP_2:6;
end;
