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

theorem Th22:
  for n being Nat holds Fint(A /\ B,n) = Fint(A,n) /\ Fint(B,n)
proof
  defpred P[Nat] means (Fint(A /\ B)).$1 = (Fint A).$1 /\ (Fint 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];
    (Fint(A /\ B)).(k+1) = Fint(A /\ B,k)^i by Def4
      .= ((Fint(A,k))^i) /\ (Fint(B,k)^i) by A2,Th10
      .= Fint(A,k+1) /\ ((Fint(B,k))^i) by Def4
      .= (Fint A).(k+1) /\ (Fint B).(k+1) by Def4;
    hence thesis;
  end;
  (Fint(A /\ B)).0 = A /\ B by Def4
    .= (Fint(A)).0 /\ B by Def4
    .= (Fint(A)).0 /\ (Fint(B)).0 by Def4;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
