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

theorem Th65:
  A1 is convergent or A2 is convergent implies lim_inf (A1 (\+\)
  A2) c= lim_inf A1 \+\ lim_inf A2
proof
A1: for A1,A2 st A1 is convergent holds lim_inf (A1 (\+\) A2) c= lim_inf A1
  \+\ lim_inf A2
  proof
    let A1,A2;
    assume
A2: A1 is convergent;
    thus 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 n1 being Nat such that
A3:   for k holds x in (A1 (\+\) A2).(n1+k) by KURATO_0:4;
A4:   now
        let k;
        x in (A1 (\+\) A2).(n1+k) by A3;
        then x in A1.(n1+k) \+\ A2.(n1+k) by Def4;
        hence x in A1.(n1+k) & not x in A2.(n1+k) or not x in A1.(n1+k) & x in
        A2.(n1+k) by XBOOLE_0:1;
      end;
      assume
A5:   not x in lim_inf A1 \+\ lim_inf A2;
      per cases by A5,XBOOLE_0:1;
      suppose
A6:     not x in lim_inf A1 & not x in lim_inf A2;
        then not x in lim_sup A1 by A2,KURATO_0:def 5;
        then consider n2 being Nat such that
A7:     for k holds not x in A1.(n2+k) by KURATO_0:5;
        consider k1 being Nat such that
A8:     not x in A2.((n1+n2)+k1) by A6,KURATO_0:4;
A9:     x in A1.(n1+(n2+k1)) & not x in A2.(n1+(n2+k1)) or not x in A1.(
        n1+(n2+k1)) & x in A2.(n1+(n2+k1)) by A4;
        not x in A1.(n2+(n1+k1)) by A7;
        hence contradiction by A8,A9;
      end;
      suppose
A10:    x in lim_inf A1 & x in lim_inf A2;
        then consider n3 being Nat such that
A11:    for k holds x in A2.(n3+k) by KURATO_0:4;
        consider n2 being Nat such that
A12:    for k holds x in A1.(n2+k) by A10,KURATO_0:4;
A13:    x in A1.(n1+(n2+n3)) & not x in A2.(n1+(n2+n3)) or not x in A1.(
        n1+(n2+n3)) & x in A2.(n1+(n2+n3)) by A4;
A14:    x in A2.(n3+(n1+n2)) by A11;
        x in A1.(n2+(n1+n3)) by A12;
        hence contradiction by A14,A13;
      end;
    end;
  end;
  assume
A15: A1 is convergent or A2 is convergent;
  per cases by A15;
  suppose
    A1 is convergent;
    hence thesis by A1;
  end;
  suppose
    A2 is convergent;
    hence thesis by A1;
  end;
end;
