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 Th8:
  for ASeq being SetSequence of Omega holds (ASeq is non-ascending
  iff Complement ASeq is non-descending)
proof
  let ASeq be SetSequence of Omega;
  thus ASeq is non-ascending implies Complement ASeq is non-descending
  proof
    assume
A1: ASeq is non-ascending;
    now
      let n,m;
      assume n <= m;
      then ASeq.m c= ASeq.n by A1;
      then (ASeq.n)` c= (ASeq.m)` by SUBSET_1:12;
      then (Complement ASeq).n c= (ASeq.m)` by PROB_1:def 2;
      hence (Complement ASeq).n c= (Complement ASeq).m by PROB_1:def 2;
    end;
    hence thesis;
  end;
  assume
A2: Complement ASeq is non-descending;
  now
    let n,m;
    assume n <= m;
    then (Complement ASeq).n c= (Complement ASeq).m by A2;
    then (ASeq.n)` c= (Complement ASeq).m by PROB_1:def 2;
    then (ASeq.n)` c= (ASeq.m)` by PROB_1:def 2;
    hence ASeq.m c= ASeq.n by SUBSET_1:12;
  end;
  hence thesis;
end;
