Direct well-definedness of the T-basis definition of the completed algebra

Prove directly that the multiplication described in the T-basis presentation of the completed Iwahori–Hecke algebra \widehat{H} (as in Theorem c_T_version_completed_algebra) is well defined, thereby allowing this T-basis description to serve as a standalone definition of \widehat{H}.

Background

The paper defines \widehat{H} via a Z-basis (Bernstein–Lusztig) presentation and later proves an equivalent T-basis description and multiplication rule.

Although the T-basis description is more natural for realizing elements as functions on G^{\ge 0}, the authors indicate they do not have a direct proof of well-definedness of this multiplication independent of the Z-basis framework.