reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;
reserve C for C_Measure of X;

theorem
  for X,C for seq being SetSequence of sigma_Field C holds Union seq in
  sigma_Field C
proof
  let X,C;
  let seq be SetSequence of sigma_Field(C);
  set Aseq = Partial_Diff_Union seq;
  rng Aseq c= sigma_Field(C) by RELAT_1:def 19;
  then reconsider Aseq9 = Aseq as sequence of sigma_Field(C) by FUNCT_2:6;
  reconsider Aseq9 as Sep_Sequence of sigma_Field(C);
  union rng Aseq9 in sigma_Field(C) by Th24;
  then Union Aseq in sigma_Field(C) by CARD_3:def 4;
  hence Union seq in sigma_Field(C) by PROB_3:36;
end;
