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 Th14:
  B is constant & the_value_of B = A implies for n holds meet {B.k
  : n <= k} = A
proof
  assume
A1: B is constant & the_value_of B = A;
  let n;
  set Y = {B.k : n <= k};
A2: now
    let x be object;
    assume x in Y;
    then ex k st x = B.k & n <= k;
    hence x = A by A1,Lm2;
  end;
  Y <> {} by Th1;
  then Y = {A} by A2,ZFMISC_1:35;
  hence thesis by SETFAM_1:10;
end;
