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

theorem
  for n being Nat holds Finf(A \/ B,n) = Finf(A,n) \/ Finf(B, n)
proof
  defpred P[Nat] means (Finf(A \/ B)).$1 = (Finf(A)).$1 \/ (Finf 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];
    (Finf(A \/ B)).(k+1) = Finf(A \/ B,k)^f by Def6
      .= Finf(A,k)^f \/ Finf(B,k)^f by A2,Th11
      .= Finf(A,k+1) \/ Finf(B,k)^f by Def6
      .= (Finf(A)).(k+1) \/ (Finf(B)).(k+1) by Def6;
    hence thesis;
  end;
  (Finf(A \/ B)).0 = A \/ B by Def6
    .= (Finf(A)).0 \/ B by Def6
    .= (Finf(A)).0 \/ (Finf(B)).0 by Def6;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
