
theorem Th60:
  for T1,T2 being non empty TopSpace st the carrier of T1 = the carrier of T2
  for T being Refinement of T1, T2
  holds INTERSECTION(the topology of T1, the topology of T2) is Basis of T
proof
  let T1,T2 be non empty TopSpace such that
A1: the carrier of T1 = the carrier of T2;
  let T be Refinement of T1, T2;
  set B1 = the topology of T1, B2 = the topology of T2;
  UniCl B1 = B1 by CANTOR_1:6;
  then reconsider B1 as Basis of T1 by Th22;
  UniCl B2 = B2 by CANTOR_1:6;
  then reconsider B2 as Basis of T2 by Th22;
A2: B1 \/ B2 \/ INTERSECTION(B1,B2) is Basis of T by Th59;
A3: the carrier of T1 in B1 by PRE_TOPC:def 1;
A4: the carrier of T2 in B2 by PRE_TOPC:def 1;
A5: B1 c= INTERSECTION(B1,B2)
  proof
    let a be object;
    reconsider aa=a as set by TARSKI:1;
    assume
A6: a in B1;
    then aa /\ the carrier of T1 in INTERSECTION(B1,B2)
       by A1,A4,SETFAM_1:def 5;
    hence thesis by A6,XBOOLE_1:28;
  end;
  B2 c= INTERSECTION(B1,B2)
  proof
    let a be object;
    reconsider aa=a as set by TARSKI:1;
    assume
A7: a in B2;
    then aa /\ the carrier of T2 in INTERSECTION(B1,B2)
    by A1,A3,SETFAM_1:def 5;
    hence thesis by A7,XBOOLE_1:28;
  end;
  hence thesis by A2,A5,XBOOLE_1:8,12;
end;
