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 Th40:
 for n being Nat holds
  B is non-descending implies B.n c= (superior_setsequence(B)).(n+ 1)
proof let n be Nat;
  assume B is non-descending;
  then
A1: B.n c= B.(n+1) by PROB_2:7;
  B.n c= union {B.k : n+1 <= k}
  proof
    let x be object;
A2: B.(n+1) in {B.k : n+1 <= k};
    assume x in B.n;
    hence thesis by A1,A2,TARSKI:def 4;
  end;
  hence thesis by Def3;
end;
