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;
