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

theorem Th42:
  for n being Nat holds Fdfl(A /\ B,n) = Fdfl(A,n) /\ Fdfl(B,n)
proof
  defpred P[Nat] means (Fdfl(A /\ B)).$1= (Fdfl A).$1 /\ (Fdfl B).
  $1;
  let n be Nat;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A2: P[k];
    (Fdfl(A /\ B)).(k+1) = Fdfl(A /\ B,k)^d by Def8
      .= ((Fdfl(A,k))^d) /\ ((Fdfl(B,k))^d) by A2,Th12
      .= Fdfl(A,k+1) /\ ((Fdfl(B,k))^d) by Def8
      .= (Fdfl A).(k+1) /\ (Fdfl B).(k+1) by Def8;
    hence thesis;
  end;
  (Fdfl(A /\ B)).0 = A /\ B by Def8
    .= (Fdfl A).0 /\ B by Def8
    .= (Fdfl A).0 /\ (Fdfl B).0 by Def8;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
