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

theorem Th45:
  for n being Nat holds Fdfl(A,n) = Finf(A`,n)`
proof
  defpred P[Nat] means (Fdfl(A)).$1= ((Finf(A`)).In($1,NAT))`;
  let n be Nat;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    assume
A2: P[k];
    (Fdfl A).(k+1) = Fdfl(A,k)^d by Def8;
    then (Fdfl A).(k+1) = ((((Fdfl(A)).kk)`)^f)` by Th4
      .= ((Finf(A`)).In(k+1,NAT))` by A2,Def6;
    hence thesis;
  end;
  ((Finf(A`)).0)` = A`` by Def6
    .= A;
  then
A3: P[0] by Def8;
   reconsider n as Element of NAT by ORDINAL1:def 12;
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  then P[n];
  hence thesis;
end;
