reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th47:
  B is non-ascending implies (superior_setsequence B).(n+1) c= B.n
proof
  assume
A1: B is non-ascending;
  let x be object;
  assume x in (superior_setsequence(B)).(n+1);
  then consider k being Nat such that
A2: x in B.(n +1+k) by Th20;
  n+1 <= n+1+k by NAT_1:11;
  then n <= n+1+k by NAT_1:13;
  then B.(n +1+k) c= B.n by A1,PROB_1:def 4;
  hence thesis by A2;
end;
