
theorem
  for FT being filled non empty RelStr, A being Subset of FT, n,m
  being Element of NAT st n<=m holds Fint(A,m) c= Fint(A,n)
proof
  let FT be filled non empty RelStr,A be Subset of FT, n,m be Element of NAT;
  defpred P[Nat] means Fint(A,n+$1) c= Fint(A,n);
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
A2: Fint(A,n+k+1) c= Fint(A,n+k) by FINTOPO3:26;
    assume P[k];
    hence thesis by A2,XBOOLE_1:1;
  end;
  assume n<=m;
  then m-n>=0 by XREAL_1:48;
  then
A3: n+(m-'n)=n+(m-n) by XREAL_0:def 2
    .=m;
A4: P[0];
  for m2 being Nat holds P[m2] from NAT_1:sch 2(A4,A1);
  hence thesis by A3;
end;
