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 Th28:
  for B,n holds {B.k: n <= k} is Subset-Family of X
proof
  let B,n;
  set Y1 = {B.k: n <= k};
  Y1 c= bool X
  proof
    let x be object;
    assume x in Y1;
    then consider k such that
A1:   x = B.k & n <= k;
     reconsider k as Element of NAT by ORDINAL1:def 12;
     x = B.k by A1;
    hence thesis;
  end;
  hence thesis;
end;
