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 Th29:
  for X being set, S be SigmaField of X, M be sigma_Measure of S,
  SSets being SetSequence of S st SSets is non-descending holds M*SSets is
  non-decreasing
proof
  let X be set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let SSets be SetSequence of S;
A1: dom(M*SSets) = NAT by FUNCT_2:def 1;
  assume
A2: SSets is non-descending;
  now
    let n,m be Nat;
A3: n in NAT & m in NAT by ORDINAL1:def 12;
A4: (M*SSets).m = M.(SSets.m) by A1,FUNCT_1:12,A3;
    assume n <= m;
    then
A5: SSets.n c= SSets.m by A2,PROB_1:def 5;
    rng SSets c= S & (M*SSets).n = M.(SSets.n) by A1,FUNCT_1:12,A3;
    hence (M*SSets).n <= (M*SSets).m by A5,A4,MEASURE1:31;
  end;
  then for n,m be Nat st m<=n holds (M*SSets).m<=(M*SSets).n;
  hence M*SSets is non-decreasing by RINFSUP2:7;
end;
