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

theorem Th79:
  A1 is convergent implies lim_inf (A (\) A1) = A \ lim_inf A1
proof
  assume
A1: A1 is convergent;
  thus lim_inf (A (\) A1) c= A \ lim_inf A1 by Th76;
  thus A \ lim_inf A1 c= lim_inf (A (\) A1)
  proof
    let x be object;
    assume
A2: x in A \ lim_inf A1;
    then x in A \ lim_sup A1 by A1,KURATO_0:def 5;
    then not x in lim_sup A1 by XBOOLE_0:def 5;
    then consider n1 being Nat such that
A3: for k holds not x in A1.(n1+k) by KURATO_0:5;
    assume
A4: not x in lim_inf (A (\) A1);
A5: for n holds not x in A or ex k st x in A1.(n+k)
    proof
      let n;
      consider k such that
A6:   not x in (A (\) A1).(n+k) by A4,KURATO_0:4;
      not x in A \ A1.(n+k) by A6,Def7;
      then not ( x in A & not x in A1.(n+k)) by XBOOLE_0:def 5;
      hence thesis;
    end;
    per cases by A5;
    suppose
      not x in A;
      hence contradiction by A2,XBOOLE_0:def 5;
    end;
    suppose
      ex k st x in A1.(n1+k);
      hence contradiction by A3;
    end;
  end;
end;
