let X be RealNormSpace; :: thesis: for U being Subset of (TopSpaceNorm X)
for Ut being Subset of (LinearTopSpaceNorm X)
for x being Point of (TopSpaceNorm X)
for xt being Point of (LinearTopSpaceNorm X) st U = Ut & x = xt holds
( U is a_neighborhood of x iff Ut is a_neighborhood of xt )
let U be Subset of (TopSpaceNorm X); :: thesis: for Ut being Subset of (LinearTopSpaceNorm X)
for x being Point of (TopSpaceNorm X)
for xt being Point of (LinearTopSpaceNorm X) st U = Ut & x = xt holds
( U is a_neighborhood of x iff Ut is a_neighborhood of xt )
let Ut be Subset of (LinearTopSpaceNorm X); :: thesis: for x being Point of (TopSpaceNorm X)
for xt being Point of (LinearTopSpaceNorm X) st U = Ut & x = xt holds
( U is a_neighborhood of x iff Ut is a_neighborhood of xt )
let x be Point of (TopSpaceNorm X); :: thesis: for xt being Point of (LinearTopSpaceNorm X) st U = Ut & x = xt holds
( U is a_neighborhood of x iff Ut is a_neighborhood of xt )
let xt be Point of (LinearTopSpaceNorm X); :: thesis: ( U = Ut & x = xt implies ( U is a_neighborhood of x iff Ut is a_neighborhood of xt ) )
assume that
A1: U = Ut and
A2: x = xt ; :: thesis: ( U is a_neighborhood of x iff Ut is a_neighborhood of xt )

hereby :: thesis: ( Ut is a_neighborhood of xt implies U is a_neighborhood of x )

assume U is a_neighborhood of x ; :: thesis: Ut is a_neighborhood of xt
then consider V being 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; :: thesis: verum

end;

assume Ut is a_neighborhood of xt ; :: thesis: U is a_neighborhood of x
then consider Vt being 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 U is a_neighborhood of x by A2, A8, CONNSP_2:6; :: thesis: verum