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

theorem Th78:
  lim_inf (A (\+\) A1) c= A \+\ lim_inf A1
proof
  let x be object;
  assume x in lim_inf (A (\+\) A1);
  then consider n1 being Nat such that
A1: for k holds x in (A (\+\) A1).(n1+k) by KURATO_0:4;
A2: now
    let k;
    x in (A (\+\) A1).(n1+k) by A1;
    then x in A \+\ A1.(n1+k) by Def9;
    hence x in A & not x in A1.(n1+k) or not x in A & x in A1.(n1+k) by
XBOOLE_0:1;
  end;
  assume not x in A \+\ lim_inf A1;
  then
A3: not x in A & not x in lim_inf A1 or x in lim_inf A1 & x in A by XBOOLE_0:1;
  per cases by A3,KURATO_0:4;
  suppose
A4: not x in A & not ex n st for k holds x in A1.(n+k);
    then ex k1 being Nat st not x in A1.(n1+k1);
    hence contradiction by A2,A4;
  end;
  suppose
A5: x in A & ex n st for k holds x in A1.(n+k);
    then consider n2 being Nat such that
A6: for k holds x in A1.(n2+k);
    x in A1.(n2+n1) by A6;
    hence contradiction by A2,A5;
  end;
end;
