reserve X for RealNormSpace;

theorem Th17:
  for X be RealNormSpace, U be Subset of X, Ut be Subset of
TopSpaceNorm X, x be Point of X, xt be Point of TopSpaceNorm X st U = Ut & x =
  xt holds U is Neighbourhood of x iff Ut is a_neighborhood of xt
proof
  let X be RealNormSpace, U be Subset of X, Ut be Subset of TopSpaceNorm X, x
  be Point of X, xt be Point of TopSpaceNorm X;
  assume that
A1: U=Ut and
A2: x=xt;
A3: now
    assume U is Neighbourhood of x;
    then consider r be Real such that
A4: r > 0 and
A5: {y where y is Point of X: ||.y-x.|| < r} c= U by NFCONT_1:def 1;
    now
      let s be object;
      assume s in {y where y is Point of X: ||.y-x.|| < r};
      then ex z be Point of X st s = z & ||.z-x.|| < r;
      hence s in the carrier of X;
    end;
    then reconsider Vt={y where y is Point of X: ||.y-x.|| < r} as Subset of
    TopSpaceNorm X by TARSKI:def 3;
    Vt={y where y is Point of X: ||.x-y.|| < r} by Lm5;
    then
A6: Vt is open by Th8;
    ||.x-x.|| =0 by NORMSP_1:6;
    then xt in Vt by A2,A4;
    hence Ut is a_neighborhood of xt by A1,A5,A6,CONNSP_2:6;
  end;
  now
    assume Ut is a_neighborhood of xt;
    then consider Vt being Subset of TopSpaceNorm X such that
A7: Vt is open and
A8: Vt c= Ut and
A9: xt in Vt by CONNSP_2:6;
    consider r be Real such that
A10: r > 0 and
A11: {y where y is Point of X: ||.x-y.|| < r} c= Vt by A2,A7,A9,Th7;
A12: {y where y is Point of X: ||.x-y.|| < r} = {y where y is Point of X:
    ||.y-x.|| < r} by Lm5;
    {y where y is Point of X: ||.x-y.|| < r} c=U by A1,A8,A11;
    hence U is Neighbourhood of x by A10,A12,NFCONT_1:def 1;
  end;
  hence thesis by A3;
end;
