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

theorem Th63:
  A1 is convergent or A2 is convergent implies lim_inf (A1 (\/) A2
  ) = lim_inf A1 \/ lim_inf A2
proof
A1: for A1,A2 st A1 is convergent holds lim_inf (A1 (\/) A2) = 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 n;
        x in (A1 (\/) A2).(n1+n) by A3;
        then x in A1.(n1+n) \/ A2.(n1+n) by Def2;
        hence x in A1.(n1+n) or x in A2.(n1+n) by XBOOLE_0:def 3;
      end;
      x in lim_inf A1 \/ lim_inf A2
      proof
        assume
A5:     not x in lim_inf A1 \/ lim_inf A2;
        then not x in lim_inf A1 by XBOOLE_0:def 3;
        then not x in lim_sup A1 by A2,KURATO_0:def 5;
        then consider n2 being Nat such that
A6:     for k holds not x in A1.(n2+k) by KURATO_0:5;
        not x in lim_inf A2 by A5,XBOOLE_0:def 3;
        then consider k1 being Nat such that
A7:     not x in A2.((n1+n2)+k1) by KURATO_0:4;
        not x in A1.(n2+(n1+k1)) by A6;
        then not x in A1.(n1+(n2+k1));
        hence contradiction by A4,A7;
      end;
      hence thesis;
    end;
    thus thesis by Th61;
  end;
  assume
A8: A1 is convergent or A2 is convergent;
  per cases by A8;
  suppose
    A1 is convergent;
    hence thesis by A1;
  end;
  suppose
    A2 is convergent;
    hence thesis by A1;
  end;
end;
