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 Th41:
 for n being Nat holds
  B is non-descending implies (superior_setsequence B).n = (
  superior_setsequence B).(n+1)
proof let n be Nat;
  assume B is non-descending;
  then (superior_setsequence B).(n+1) \/ B.n = (superior_setsequence B).(n+1)
  by Th40,XBOOLE_1:12;
  hence thesis by Th22;
end;
