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

theorem Th76:
  lim_inf (A (\) A1) c= A \ lim_inf A1
proof
  let x be object;
  assume x in lim_inf (A (\) A1);
  then consider n such that
A1: for k holds x in (A (\) A1).(n+k) by KURATO_0:4;
A2: now
    let k;
    x in (A (\) A1).(n+k) by A1;
    then x in A \ A1.(n+k) by Def7;
    hence x in A & not x in A1.(n+k) by XBOOLE_0:def 5;
  end;
  not x in lim_inf A1
  proof
    assume x in lim_inf A1;
    then consider n1 being Nat such that
A3: for k holds x in A1.(n1+k) by KURATO_0:4;
    x in A1.(n1+n) by A3;
    hence contradiction by A2;
  end;
  hence thesis by A2,XBOOLE_0:def 5;
end;
