
theorem
  for FT1 being non empty RelStr, FT2 being filled non empty RelStr, n
being Element of NAT, f being Function of FT1, FT2 st f is_continuous 0 holds f
  is_continuous n
proof
  let FT1 be non empty RelStr, FT2 be filled non empty RelStr,n be Element
  of NAT, f be Function of FT1, FT2;
  assume
A1: f is_continuous 0;
  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
A2: y=f.x;
    U_FT y =U_FT(y,0) & U_FT(y,n)=Finf((U_FT y),n) by FINTOPO3:47,def 10;
    then
A3: U_FT(y,0) c= U_FT(y,n) by FINTOPO3:36;
    f.:( U_FT(x,0)) c= U_FT(y,0) by A1,A2;
    hence thesis by A3;
  end;
  hence thesis;
end;
