
theorem
  for X being set, A, B being SetSequence of X, C being Subset of X st (
  for n being Nat holds B.n = C \+\ A.n) holds C \+\ lim_inf A c=
  lim_sup B
proof
  let X be set, A, B be SetSequence of X, C be Subset of X;
  assume
A1: for n being Nat holds B.n = C \+\ A.n;
  let x be object;
  assume
A2: x in C \+\ lim_inf A;
  per cases by A2,XBOOLE_0:1;
  suppose
A3: x in C & not x in lim_inf A;
    for n being Nat ex k being Nat st x in B.(n+k)
    proof
      let n be Nat;
      consider k being Nat such that
A4:   not x in A.(n+k) by A3,Th4;
      take k;
      x in C \+\ A.(n+k) by A3,A4,XBOOLE_0:1;
      hence thesis by A1;
    end;
    hence thesis by Th5;
  end;
  suppose
A5: not x in C & x in lim_inf A;
    then consider n being Nat such that
A6: for k being Nat holds x in A.(n+k) by Th4;
    for m being Nat ex k being Nat st x in B.(m+k)
    proof
      let m be Nat;
      take k = n;
      x in A.(m+k) by A6;
      then x in C \+\ A.(m+k) by A5,XBOOLE_0:1;
      hence thesis by A1;
    end;
    hence thesis by Th5;
  end;
end;
