reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th35:
  (for m be Nat holds F.m is_simple_func_in S) implies F is
  additive & (Partial_Sums F).n is_simple_func_in S
proof
  defpred P[Nat] means (Partial_Sums F).$1 is_simple_func_in S;
  assume
A1: for m be Nat holds F.m is_simple_func_in S;
  hereby
    let n,m be Nat;
    assume n <> m;
    F.n is_simple_func_in S by A1;
    then F.n is without+infty by MESFUNC5:14;
    hence for x be set st x in dom(F.n) /\ dom(F.m) holds (F.n).x <> +infty or
    (F.m).x <> -infty;
  end;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    F.(k+1) is_simple_func_in S by A1;
    then (Partial_Sums F).k + F.(k+1) is_simple_func_in S by A3,MESFUNC5:38;
    hence thesis by Def4;
  end;
  (Partial_Sums F).0 = F.0 by Def4;
  then
A4: P[ 0 ] by A1;
  for k be Nat holds P[k] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
