reserve
  r,s,r0,s0,t for Real;

theorem Th5:
  for A,B being compact Subset of REAL holds A /\ B is compact
proof
  let A,B be compact Subset of REAL;
  let s1 be Real_Sequence such that
A1: rng s1 c= A /\ B;
A2: A /\ B c= B by XBOOLE_1:17;
  A /\ B c= A by XBOOLE_1:17;
  then rng s1 c= A by A1;
  then consider s2 being Real_Sequence such that
A3: s2 is subsequence of s1 and
A4: s2 is convergent and
A5: lim s2 in A by RCOMP_1:def 3;
  rng s2 c= rng s1 by A3,VALUED_0:21;
  then rng s2 c= A /\ B by A1;
  then rng s2 c= B by A2;
  then consider s3 being Real_Sequence such that
A6: s3 is subsequence of s2 and
A7: s3 is convergent and
A8: lim s3 in B by RCOMP_1:def 3;
  take s3;
  thus s3 is subsequence of s1 by A3,A6,VALUED_0:20;
  thus s3 is convergent by A7;
  lim s3 = lim s2 by A4,A6,SEQ_4:17;
  hence thesis by A5,A8,XBOOLE_0:def 4;
end;
