reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, O being local_base of X, P being
Subset-Family of X st P = {a+U where a is Point of X, U is Subset of X: U in O}
  holds P is basis of X
proof
  let X be LinearTopSpace, O be basis of 0.X, P be Subset-Family of X such
  that
A1: P = {a+U where a is Point of X, U is Subset of X: U in O};
  let p be Point of X;
  let A be a_neighborhood of p;
  p in Int(A) by CONNSP_2:def 1;
  then -p+Int(A) is a_neighborhood of 0.X by Th9,CONNSP_2:3;
  then consider V being a_neighborhood of 0.X such that
A2: V in O and
A3: V c= -p+Int(A) by YELLOW13:def 2;
  take U = p+V;
  0.X in Int(V) by CONNSP_2:def 1;
  then p+0.X in p+Int(V) by Lm1;
  then
A4: p in p+Int(V);
  p+Int(V) c= Int(U) by Th37;
  hence U is a_neighborhood of p by A4,CONNSP_2:def 1;
  thus U in P by A1,A2;
  U c= p+(-p+Int(A)) by A3,Th8;
  then U c= p+-p+Int(A) by Th6;
  then U c= 0.X+Int(A) by RLVECT_1:5;
  then Int(A) c= A & U c= Int(A) by Th5,TOPS_1:16;
  hence thesis;
end;
