  In this paper the author discuss the relation between Lagrangian Floer
homology and Gauge-theory (Donaldson theory) Floer homology. It can be regarded
as a version of Atiyah-Floer type conjecture in the case of $SO(3)$-bundle with
non-trivial second Stiefel-Whitney class. This is a first of a series of
papers, where we describe the main results and geometric and algebraic parts of
the proof. The half of analytic detail was in [Fu5] which was published in
1998. The other half will appear in subsequent papers.
