reserve T for non empty RelStr,
  A,B for Subset of T,
  x,x2,y,z for Element of T;

theorem
  T is filled implies for n being Nat holds A c= Finf(A,n)
proof
  defpred P[Nat] means A c= (Finf A).$1;
  assume
A1: T is filled;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    then A^f c= Finf(A,k)^f by Th5;
    then
A3: A^f c= Finf(A,k+1) by Def6;
    A c= A^f by A1,Th1;
    hence thesis by A3,XBOOLE_1:1;
  end;
  let n be Nat;
A4: P[0] by Def6;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
