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

theorem Th17:
  for n being Nat holds Fcl(A \/ B,n) = Fcl(A,n) \/ Fcl (B,n)
proof
  let n be Nat;
  for n being Nat holds (Fcl(A \/ B)).n = (Fcl A).n \/ (Fcl B). n
  proof
    defpred P[Nat] means (Fcl(A \/ B)).$1 = (Fcl A).$1 \/ (Fcl B).
    $1;
A1: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A2:   P[k];
      (Fcl(A \/ B)).(k+1) = Fcl(A \/ B,k)^b by Def2
        .= Fcl(A,k)^b \/ Fcl(B,k)^b by A2,Th8
        .= Fcl(A,k+1) \/ Fcl(B,k)^b by Def2
        .= (Fcl(A)).(k+1) \/ (Fcl(B)).(k+1) by Def2;
      hence thesis;
    end;
    (Fcl(A \/ B)).0 = A \/ B by Def2
      .= (Fcl A).0 \/ B by Def2
      .= (Fcl A).0 \/ (Fcl B).0 by Def2;
    then
A3: P[0];
    for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
    hence thesis;
  end;
  hence thesis;
end;
