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

theorem Th27:
  for n being Nat holds Fint(A`,n)` = Fcl(A,n)
proof
  defpred P[Nat] means Fint(A`,$1)` = Fcl(A,$1);
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A2: P[k];
    Fcl(A,k+1) = Fcl(A,k)^b by Def2
      .= ((Fint(A`,k)``)^i)` by A2,FIN_TOPO:16
      .= Fint(A`,k+1)` by Def4;
    hence thesis;
  end;
  Fint(A`,0)` = A`` by Def4
    .= Fcl(A,0) by Def2;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
