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
  FSets is non-ascending implies M*FSets is non-increasing
proof
A1: dom(M*FSets) = NAT by FUNCT_2:def 1;
  assume
A2: FSets is non-ascending;
  now
    let n,m be Nat;
A3: n in NAT & m in NAT by ORDINAL1:def 12;
    assume m <= n;
    then
A4: FSets.n c= FSets.m by A2,PROB_1:def 4;
    (M*FSets).n = M.(FSets.n) & (M*FSets).m = M.(FSets.m) by A1,FUNCT_1:12,A3;
    hence (M*FSets).n <= (M*FSets).m by A4,MEASURE1:8;
  end;
  hence M*FSets is non-increasing by RINFSUP2:7;
end;
