
theorem
  for T being set, F being SetSequence of T st F is non-descending holds
  lim_sup F = Union F
proof
  let T be set, F be SetSequence of T;
  assume
A1: F is non-descending;
  thus lim_sup F c= Union F by Th8;
  let x be object;
  assume x in Union F;
  then consider n being Nat such that
A2: x in F.n by PROB_1:12;
  assume not x in lim_sup F;
  then consider m being Nat such that
A3: for k being Nat holds not x in F.(m+k) by Th5;
A4: not x in F.(m+(0 qua Nat)) by A3;
  per cases;
  suppose
    n <= m;
    then F.n c= F.m by A1,PROB_1:def 5;
    hence thesis by A2,A4;
  end;
  suppose
    n > m;
    then consider h being Nat such that
A5: n = m + h by NAT_1:10;
    thus thesis by A2,A3,A5;
  end;
end;
