reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th38:
  for D being u.s.c._decomposition of X, t being Point of space D,
G being a_neighborhood of Proj D " {t} holds Proj(D).:G is a_neighborhood of t
proof
  let D be u.s.c._decomposition of X, t be Point of space D, G be
  a_neighborhood of Proj D " {t};
A1: the carrier of space D = D by Def7;
  then Proj D " {t} = t by EQREL_1:66;
  then consider W being Subset of X such that
A2: W is open and
A3: Proj D " {t} c= W and
A4: W c= G and
A5: for B being Subset of X st B in D & B meets W holds B c= W by A1,Def10;
  set Q = Proj D .:W;
A6: Proj D .: Proj D " {t} c= Q by A3,RELAT_1:123;
  union Q = proj D " Q by A1,EQREL_1:67
    .= W by A5,EQREL_1:69;
  then union Q in the topology of X by A2;
  then Q in the topology of space D by A1,Th27;
  then
A7: Q is open;
  rng Proj D = the carrier of space D by Th30;
  then {t} c= Q by A6,FUNCT_1:77;
  then
A8: t in Q by ZFMISC_1:31;
  Q c= (Proj D).:G by A4,RELAT_1:123;
  then t in Int((Proj D).:G) by A7,A8,TOPS_1:22;
  hence thesis by CONNSP_2:def 1;
end;
