I will upgrade the version 6.1 to 7.1 in CUCM. I can ask directly from one version to another.
If you change your version of CCM (i.e. go from 6.x to 7.x) you will definately need an upgrade license. This adds to your current set of licenses to upgrade them to v7.0.
If you mean by ' I bought the current license with the right to upgrade' you have UCSS, then you can order the upgrade kit which will include media and licenses from the Product Upgrade Tool (http://tools.cisco.com/gct/Upgrade/jsp/index.jsp).
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A general rule of thumb is that you need to be on at least 6.1(3) before you can upgrade directly to a 7x version of CUCM. For the specifics, check Walter's link to verify your starting version and the target version you'd like to upgrade to. Good link, Walter - 5 to you.
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In the follow Link you can double check if this direct upgrade can be done. I am sorry for not tell you yes or no, but you didn't provide us the specific versions .
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Cisco UCCX Specialist