reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;
reserve X,Y for non empty TopSpace;

theorem
  (for X being non empty TopSpace, f being Function of X,Y holds f is
  continuous ) implies Y is anti-discrete
proof
  set X = ADTS(the carrier of Y);
A1: X = TopStruct(#the carrier of Y, cobool the carrier of Y#) by TEX_1:def 3;
  then reconsider f = id (the carrier of Y) as Function of X,Y;
  assume for X being non empty TopSpace, f being Function of X,Y holds f is
  continuous;
  then
A2: f is continuous;
  for A being Subset of Y st A is closed holds A = {} or A = the carrier of Y
  proof
    let A be Subset of Y;
    reconsider B = A as Subset of X by A1;
A3: f"A = B by FUNCT_2:94;
    assume A is closed;
    then B is closed by A2,A3;
    hence thesis by A1,TDLAT_3:19;
  end;
  hence thesis by TDLAT_3:19;
end;
