HTTPS to IIS from browsers

Using browsers as HTTP clients to the same site (https://putest/mirjanaTest/test.html) as SI business process did, with client authentication included.
Internet Explorer
Tools >> Internet Options >> Content >> Certificates
>> Import ...
When running https://putest/mirjanaTest/test.html we are asked for certificate by which the client will be authenticated at the server side:
Firefox
Tools >> Options >> View Certificates >> Import ...
When running https://putest/mirjanaTest/test.html we are asked for certificate by which the client will be authenticated at the server side: