reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;

theorem Th4:
  (ASeq is non-ascending & for n holds BSeq.n = ASeq.n /\ B)
  implies BSeq is non-ascending
proof
  assume that
A1: ASeq is non-ascending and
A2: for n holds BSeq.n = ASeq.n /\ B;
  thus BSeq qua Function is non-ascending
  proof
    let m,n;
    assume m<=n;
    then ASeq.n c= ASeq.m by A1;
    then ASeq.n /\ B c= ASeq.m /\ B by XBOOLE_1:26;
    then BSeq.n c= ASeq.m /\ B by A2;
    hence BSeq.n c= BSeq.m by A2;
  end;
end;
