Given a pair B, C of q-commuting, hereditary C-subalgebras of a unita
l C-algebra A, such that B boolean AND C is sigma-unital and 1 is an
element of B + C, there is an element h in A, with 0 less than or equa
l to h less than or equal to 1, such that II is strictly positive in B
and 1 - h is strictly positive in in C. Moreover, h - h(2) is strictl
y positive in in B boolean AND C.