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 Th34:
  (for n holds M.(FSets.n) < +infty) implies M.((Partial_Union
  FSets).k) < +infty
proof
  defpred P[Nat] means M.((Partial_Union FSets).$1) < +infty;
  assume
A1: for n holds M.(FSets.n) < +infty;
A2: now
    let k be Nat;
A3: In(0,REAL) <= M.((Partial_Union FSets).k) by SUPINF_2:51;
    M.(FSets.(k+1)) < +infty & In(0,REAL) <= M.(FSets.(k+1))
         by A1,SUPINF_2:51;
    then
A4: M.(FSets.(k+1)) in REAL by XXREAL_0:10;
    assume P[k];
    then M.((Partial_Union FSets).k) in REAL by A3,XXREAL_0:10;
    then
A5: M.((Partial_Union FSets).k) + M.(FSets.(k+1)) in REAL by A4,XREAL_0:def 1;
    Partial_Union FSets is Set_Sequence of F by Th15;
    then
A6: (Partial_Union FSets).k in F by Def2;
    (Partial_Union FSets).(k+1) = (Partial_Union FSets).k \/ FSets.(k+1)
    by PROB_3:def 2;
    then M.((Partial_Union FSets).(k+1)) <= M.((Partial_Union FSets).k) + M.(
    FSets.(k+1)) by A6,MEASURE1:10;
    hence P[k+1] by A5,XXREAL_0:9,13;
  end;
  (Partial_Union FSets).0 = FSets.0 by PROB_3:def 2;
  then
A7: P[0] by A1;
  for k being Nat holds P[k] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
