reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th24:
  for T1,T2 being non empty TopSpace for B1 being
  Neighborhood_System of T1 for B2 being Neighborhood_System of T2 st B1 = B2
  holds the TopStruct of T1 = the TopStruct of T2
proof
  let T1,T2 be non empty TopSpace;
  let B1 be Neighborhood_System of T1;
  let B2 be Neighborhood_System of T2;
A1: dom B1 = the carrier of T1 by PARTFUN1:def 2;
A2: dom B2 = the carrier of T2 by PARTFUN1:def 2;
A3: UniCl Union B2 = the topology of T2 by YELLOW_9:22;
A4: UniCl Union B1 = the topology of T1 by YELLOW_9:22;
  assume B1 = B2;
  hence thesis by A4,A3,A1,A2;
end;
