  We prove a new general multiplicity estimate applicable to sets of functions
without any assumption on algebraic independence. The multiplicity estimates
are commonly used in determining measures of algebraic independence of values
of functions, for instance within the context of Mahler's method. For this
reason, our result provides an important tool for the proofs of algebraic
independence of complex numbers. At the same time, these estimates can be
considered as a measure of algebraic independence of functions themselves.
Hence our result provides, under some conditions, the measure of algebraic
independence of elements in ${\bf F}_q[[T]]$, where ${\bf F}_q$ denotes a
finite field.
