First three lines between begin...end are from original code, which I installed to Lazarus through OPM. I added two last lines
FConnection.UserName := <your user name>;
FConnection.Password := <your password>;
In the documentation author mention, that connection through proxy server is supported by the adding additional parameters to Address string:
https://wiki.freepascal.org/Web_Service_Toolkit#HTTP_Proxy_Parameters.
I try to find out the place in the code, where this string is parsed to set ProxyServer, ProxyPort, ProxyUsername, ProxyPassword parameters for the connection and add my keys UserName and Password to set Basic HTTP Authentification parameters for the connection, but I'm failed and I added two lines to the constructor
THTTPTransport.Create().