reserve a,b,c for set;

theorem
  for T being non empty TopSpace for x,y being Point of T for B1 being
Basis of x, B2 being Basis of y for U being set st x in U & U in B2 ex V being
  open Subset of T st V in B1 & V c= U
proof
  let T be non empty TopSpace;
  let x,y be Point of T;
  let B1 be Basis of x;
  let B2 be Basis of y;
  let U be set;
  assume that
A1: x in U and
A2: U in B2;
  U is open Subset of T by A2,YELLOW_8:12;
  then consider V being Subset of T such that
A3: V in B1 and
A4: V c= U by A1,YELLOW_8:13;
  V is open by A3,YELLOW_8:12;
  hence thesis by A3,A4;
end;
