Example code : Appending a line of text to an existing file
var
  myFile : TextFile;
  text   : string;

begin
  // Try to open the Test.txt file for writing to
  AssignFile(myFile, 'Test.txt');
  ReWrite(myFile);

  // Write a couple of well known words to this file
  WriteLn(myFile, 'Hello');
  WriteLn(myFile, 'World');

  // Close the file
  CloseFile(myFile);

  // Reopen to append a final line to the file
  Append(myFile);

  // Write this final line
  WriteLn(myFile, 'Final line added');

  // Close the file
  CloseFile(myFile);

  // Reopen the file for reading
  Reset(myFile);

  // Display the file contents
  while not Eof(myFile) do
  begin
    ReadLn(myFile, text);
    ShowMessage(text);
  end;

  // Close the file for the last time
  CloseFile(myFile);
end;

Another way:

var sl : TStringList;
Begin
    sl := TStringList.Create;
    if FileExists(FileName) then
        sl.LoadFromFile(FileName);
    sl.AddStrings(OtherStringList);
    sl.SaveToFile(FileName);
    sl.free;
end;
 
   

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it's not too optimal :) consider the case when you want to just add lines. like a logging system. or any other case in which you don't care what;s in the file, you jsut need to add content. if the file grows to 100 MB, your solution will be very slow and will eat up memory, whereas a simple append will not care about what is in the file and will have the same speed and memory consumption over execution ;)
plus you don't provide resource protection. if file exists but you have no rights to read it or write to it, you will hvae an error on that line which will have as a side effect a memory leak :)
otehr than that it's an interesting solution.

this post is only meant as educational :)