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 Th19:
 for n being Nat holds
  x in (inferior_setsequence B).n iff
     for k being Nat holds x in B.(n+k)
proof let n be Nat;
  reconsider nn=n as Nat;
  set Y = {B.k : nn <= k};
A1: (inferior_setsequence B).n
   = meet {B.k : n <= k} by Def2;
A2: now
    assume
A3: x in (inferior_setsequence B).n;
    now
      let k be Nat;
      B.(n + k) in Y by Th1;
      hence x in B.(n + k) by A1,A3,SETFAM_1:def 1;
    end;
    hence for k being Nat holds x in B.(n+k);
  end;
A4: Y <> {} by Th1;
  now
    assume for k holds x in B.(n+k);
    then for Z st Z in Y holds x in Z by Th3;
    hence x in (inferior_setsequence B).n by A1,A4,SETFAM_1:def 1;
  end;
  hence thesis by A2;
end;
