reserve a,b,c for set;

theorem
  for T being non empty TopSpace for x being Point of T for B being
Basis of x for U1,U2 being set st U1 in B & U2 in B ex V being open Subset of T
  st V in B & V c= U1 /\ U2
proof
  let T be non empty TopSpace;
  let x be Point of T;
  let B be Basis of x;
  let U1,U2 be set;
  assume that
A1: U1 in B and
A2: U2 in B;
  reconsider U1, U2 as open Subset of T by A1,A2,YELLOW_8:12;
A3: x in U2 by A2,YELLOW_8:12;
  x in U1 by A1,YELLOW_8:12;
  then x in U1/\U2 by A3,XBOOLE_0:def 4;
  then consider V being Subset of T such that
A4: V in B and
A5: V c= U1/\U2 by YELLOW_8:13;
  V is open by A4,YELLOW_8:12;
  hence thesis by A4,A5;
end;
