
theorem
  for FT1 being non empty RelStr, FT2 being filled non empty RelStr,
n0,n being Element of NAT, f being Function of FT1, FT2 st f is_continuous n0 &
  n0<=n holds f is_continuous n
proof
  let FT1 be non empty RelStr, FT2 be filled non empty RelStr,n0,n be
  Element of NAT, f be Function of FT1, FT2;
  assume that
A1: f is_continuous n0 and
A2: n0<=n;
  for x being Element of FT1,y being Element of FT2 st x in the carrier of
  FT1 & y=f.x holds f.:( U_FT(x,0)) c= U_FT(y,n)
  proof
    let x be Element of FT1,y be Element of FT2;
    assume that
    x in the carrier of FT1 and
A3: y=f.x;
    U_FT(y,n0)=Finf((U_FT y),n0) & U_FT(y,n)=Finf((U_FT y),n) by
FINTOPO3:def 10;
    then
A4: U_FT(y,n0) c= U_FT(y,n) by A2,Th1;
    f.:( U_FT(x,0)) c= U_FT(y,n0) by A1,A3;
    hence thesis by A4;
  end;
  hence thesis;
end;
