reserve X for RealNormSpace;

theorem Th25:
  for X be RealNormSpace, S be Subset-Family of TopSpaceNorm X, St
  be Subset-Family of LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be
  Point of LinearTopSpaceNorm X st S=St & x=xt holds St is Basis of xt iff S is
  Basis of x
proof
  let X be RealNormSpace, S be Subset-Family of TopSpaceNorm X, St be
  Subset-Family of LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be
  Point of LinearTopSpaceNorm X;
  assume that
A1: S=St and
A2: x=xt;
A3: Intersect S = Intersect St by A1,Def4;
  hereby
    assume
A4: St is Basis of xt;
    then St c= the topology of LinearTopSpaceNorm X by TOPS_2:64;
    then
A5: S c= the topology of TopSpaceNorm X by A1,Def4;
A6: now
      let U be Subset of TopSpaceNorm X such that
A7:   U is open and
A8:   x in U;
      reconsider Ut=U as open Subset of LinearTopSpaceNorm X by A7,Def4,Th20;
      consider Vt being Subset of LinearTopSpaceNorm X such that
A9:   Vt in St & Vt c= Ut by A2,A4,A8,YELLOW_8:def 1;
      reconsider V=Vt as Subset of TopSpaceNorm X by Def4;
      take V;
      thus V in S & V c= U by A1,A9;
    end;
    x in Intersect S by A2,A3,A4,YELLOW_8:def 1;
    hence S is Basis of x by A5,A6,TOPS_2:64,YELLOW_8:def 1;
  end;
  assume
A10: S is Basis of x;
  then S c= the topology of TopSpaceNorm X by TOPS_2:64;
  then
A11: St c= the topology of LinearTopSpaceNorm X by A1,Def4;
A12: now
    let Ut be Subset of LinearTopSpaceNorm X such that
A13: Ut is open and
A14: xt in Ut;
    reconsider U=Ut as open Subset of TopSpaceNorm X by A13,Def4,Th20;
    consider V being Subset of TopSpaceNorm X such that
A15: V in S & V c= U by A2,A10,A14,YELLOW_8:def 1;
    reconsider Vt=V as Subset of LinearTopSpaceNorm X by Def4;
    take Vt;
    thus Vt in St & Vt c= Ut by A1,A15;
  end;
  xt in Intersect St by A2,A3,A10,YELLOW_8:def 1;
  hence thesis by A11,A12,TOPS_2:64,YELLOW_8:def 1;
end;
