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