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

theorem Th62:
  lim_inf (A1 (\) A2) c= lim_inf A1 \ lim_inf A2
proof
  let x be object;
  assume x in lim_inf (A1 (\) A2);
  then consider n such that
A1: for k holds x in (A1 (\) A2).(n+k) by KURATO_0:4;
A2: now
    let k;
    x in (A1 (\) A2).(n+k) by A1;
    then x in A1.(n+k) \ A2.(n+k) by Def3;
    hence x in A1.(n+k) & not x in A2.(n+k) by XBOOLE_0:def 5;
  end;
A3: not x in lim_inf A2
  proof
    assume x in lim_inf A2;
    then consider n1 being Nat such that
A4: for k holds x in A2.(n1+k) by KURATO_0:4;
    x in A2.(n1+n) by A4;
    hence contradiction by A2;
  end;
  x in lim_inf A1 by A2,KURATO_0:4;
  hence thesis by A3,XBOOLE_0:def 5;
end;
