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

theorem Th66:
  A1 is convergent & A2 is convergent implies lim_inf (A1 (\+\) A2
  ) = lim_inf A1 \+\ lim_inf A2
proof
  assume that
A1: A1 is convergent and
A2: A2 is convergent;
  thus lim_inf (A1 (\+\) A2) c= lim_inf A1 \+\ lim_inf A2 by A1,Th65;
  thus lim_inf A1 \+\ lim_inf A2 c= lim_inf (A1 (\+\) A2)
  proof
    let x be object;
    assume
A3: x in lim_inf A1 \+\ lim_inf A2;
    per cases by A3,XBOOLE_0:1;
    suppose
A4:   x in lim_inf A1 & not x in lim_inf A2;
      then not x in lim_sup A2 by A2,KURATO_0:def 5;
      then consider n2 being Nat such that
A5:   for k holds not x in A2.(n2+k) by KURATO_0:5;
      consider n1 being Nat such that
A6:   for k holds x in A1.(n1+k) by A4,KURATO_0:4;
      now
        let k;
        x in A1.(n1+(n2+k)) & not x in A2.(n2+(n1+k)) by A6,A5;
        then x in A1.(n1+n2+k) \+\ A2.(n1+n2+k) by XBOOLE_0:1;
        hence x in (A1 (\+\) A2). (n1+n2+k) by Def4;
      end;
      hence thesis by KURATO_0:4;
    end;
    suppose
A7:   not x in lim_inf A1 & x in lim_inf A2;
      then not x in lim_sup A1 by A1,KURATO_0:def 5;
      then consider n3 being Nat such that
A8:   for k holds not x in A1.(n3+k) by KURATO_0:5;
      consider n4 being Nat such that
A9:   for k holds x in A2.(n4+k) by A7,KURATO_0:4;
      now
        let k;
        ( not x in A1.(n3+(n4+k)))& x in A2.(n4+(n3+k)) by A8,A9;
        then x in A1.(n3+n4+k) \+\ A2.(n3+n4+k) by XBOOLE_0:1;
        hence x in (A1 (\+\) A2). (n3+n4+k) by Def4;
      end;
      hence thesis by KURATO_0:4;
    end;
  end;
end;
