reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th7:
  ASeq is non-ascending implies P * ASeq is non-increasing
proof
A1: dom (P * ASeq) = NAT by SEQ_1:1;
  assume
A2: ASeq is non-ascending;
  now
    let n,m be Nat;
    assume n <= m;
    then
A3: ASeq.m c= ASeq.n by A2,PROB_1:def 4;
     reconsider nn=n, mm=m as Element of NAT by ORDINAL1:def 12;
    (P * ASeq).nn = P.(ASeq.nn) & (P * ASeq).mm = P.(ASeq.mm)
          by A1,FUNCT_1:12;
    hence (P * ASeq).m <= (P * ASeq).n by A3,PROB_1:34;
  end;
  hence thesis by SEQM_3:8;
end;
