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 Th50:
 for n being Nat holds
  B is non-ascending implies (inferior_setsequence(B)).(n+1) c= B. n
proof let n be Nat;
  set Y = {B.k : n+1 <= k};
  assume B is non-ascending;
  then
A1: B.(n+1) c= B.n by PROB_2:6;
A2: B.(n+1) in Y;
A3: now
    let x be object;
    assume x in meet Y;
    then x in B.(n+1) by A2,SETFAM_1:def 1;
    hence x in B.n by A1;
  end;
  (inferior_setsequence(B)).(n+1) = meet {B.k : n+1 <= k} by Def2;
  hence thesis by A3;
end;
