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
  for X being set, S being SetSequence of X holds S is non-descending
  iff for n holds S.n c= S.(n+1)
proof
  let X be set, S be SetSequence of X;
  thus S is non-descending implies for n holds S.n c= S.(n+1)
  by NAT_1:11;
  assume
A1: for n holds S.n c= S.(n+1);
  now
    let n,m such that
A2: n <= m;
A3: now
      defpred P[Nat] means S.n c= S.(n+$1);
A4:   for k st P[k] holds P[k+1]
      proof
        let k such that
A5:     S.n c= S.(n+k);
        S.(n+k) c= S.(n+k+1) by A1;
        hence thesis by A5,XBOOLE_1:1;
      end;
A6:   P[0];
      thus for k holds P[k] from NAT_1:sch 2(A6,A4);
    end;
    consider k being Nat such that
A7: m = n + k by A2,NAT_1:10;
    thus S.n c= S.m by A3,A7;
  end;
  hence thesis;
end;
