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 Th51:
 for n being Nat holds
  B is non-ascending implies (inferior_setsequence(B)).n = (
  inferior_setsequence(B)).(n+1)
proof let n be Nat;
  assume B is non-ascending;
  then
  (inferior_setsequence(B)).(n+1) /\ B.n = (inferior_setsequence(B)).(n+1)
  by Th50,XBOOLE_1:28;
  hence thesis by Th21;
end;
