reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th71:
  A1 is convergent or A2 is convergent implies lim_sup (A1 (/\) A2
  ) = lim_sup A1 /\ lim_sup A2
proof
A1: for A1,A2 st A1 is convergent holds lim_sup (A1 (/\) A2) = lim_sup A1 /\
  lim_sup A2
  proof
    let A1,A2;
    assume
A2: A1 is convergent;
    thus lim_sup (A1 (/\) A2) c= lim_sup A1 /\ lim_sup A2 by Th67;
    thus lim_sup A1 /\ lim_sup A2 c= lim_sup (A1 (/\) A2)
    proof
      let x be object;
      assume
A3:   x in lim_sup A1 /\ lim_sup A2;
      then x in lim_sup A1 by XBOOLE_0:def 4;
      then x in lim_inf A1 by A2,KURATO_0:def 5;
      then consider n1 being Nat such that
A4:   for k holds x in A1.(n1+k) by KURATO_0:4;
A5:   x in lim_sup A2 by A3,XBOOLE_0:def 4;
      now
        let n;
        consider k1 being Nat such that
A6:     x in A2.((n+n1)+k1) by A5,KURATO_0:5;
        x in A1.(n1+(n+k1)) by A4;
        then x in A1.(n+(n1+k1)) /\ A2.(n+(n1+k1)) by A6,XBOOLE_0:def 4;
        then x in (A1 (/\) A2).(n+(n1+k1)) by Def1;
        hence ex k st x in (A1 (/\) A2).(n+k);
      end;
      hence thesis by KURATO_0:5;
    end;
  end;
  assume
A7: A1 is convergent or A2 is convergent;
  per cases by A7;
  suppose
    A1 is convergent;
    hence thesis by A1;
  end;
  suppose
    A2 is convergent;
    hence thesis by A1;
  end;
end;
