
theorem
  for X being LinearTopSpace,
      O being local_base of X,
      a being Point of X,
      P being Subset-Family of X st P = {a+U where U is Subset of X: U in O}
  holds P is basis of a
  proof
    let X be LinearTopSpace, O be basis of 0.X,
    a be Point of X,
    P be Subset-Family of X such that
A1: P = {a+U where U is Subset of X: U in O};
    let A be a_neighborhood of a;
    a in Int(A) by CONNSP_2:def 1;
    then (-a) + Int(A) is a_neighborhood of 0.X by RLTOPSP1:9,CONNSP_2:3;
    then consider V being a_neighborhood of 0.X such that
A2: V in O and
A3: V c= -a+Int(A) by YELLOW13:def 2;
    take U = a+V;
A4: a+0.X in a+Int(V) by Lm1,CONNSP_2:def 1;
    a+Int(V) c= Int(U) by RLTOPSP1:37;
    hence U is a_neighborhood of a by A4,CONNSP_2:def 1;
    thus U in P by A1,A2;
    U c= a+(-a+Int(A)) by A3,RLTOPSP1:8;
    then U c= a+(-a)+Int(A) by RLTOPSP1:6;
    then U c= 0.X+Int(A) by RLVECT_1:5;
    then Int(A) c= A & U c= Int(A) by RLTOPSP1:5,TOPS_1:16;
    hence thesis;
  end;
