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;

theorem Th16:
  Partial_Diff_Union FSets is Set_Sequence of F
proof
  defpred P[Nat] means (Partial_Diff_Union FSets).$1 in F;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    Partial_Union FSets is Set_Sequence of F by Th15;
    then
A2: (Partial_Union FSets).k in F by Def2;
    (Partial_Diff_Union FSets).(k+1) = FSets.(k+1) \ (Partial_Union FSets)
    .k by PROB_3:def 3;
    hence P[k+1] by A2,PROB_1:6;
  end;
  (Partial_Diff_Union FSets).0 = FSets.0 by PROB_3:def 3;
  then
A3: P[0];
  for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis by Def2;
end;
