As　homogeneous　wavelet　bi-frames　derived　from　a　pair　of　refinable　functions,　nonhomogeneous　wavelet　bi-frames　admit　fast　wavelet　transform,　which　is　one　of　main　concerns　in　wavelet　analysis.　Simultaneously,　for　a　nonhomogeneous　wavelet　bi-frame　in　a　pair　of　dual　spaces　(H-s　(R-d),　H-s　(R-d))　with　s　not　equal　0,　smoothness　and　vanishing　moment　requirements　are　separated　from　each　other,　that　is,　one　system　is　for　smoothness　and　the　other　for　vanishing　moments.　This　gives　us　more　flexibility　to　construct　nonhomogeneous　wavelet　bi-frames　than　in　L-2(R-d).　However,　the　requirement　of　the　Bessel　property　of　affine　systems　is　always　needed.　For　refinable-function　based　wavelet　bi-frames,　such　a　requirement　is　imposed　on　refinable　functions　in　dierent　ways.　This　is　not　what　we　expect.　So　a　natural　question　involves　what　we　are　expecting　from　a　pair　of　general　refinable　functions.　For　this　purpose,　we　introduce　the　notion　of　the　reducing　subspace　of　a　Sobolev　space　and　the　notion　of　the　weak　nonhomogeneous　wavelet　bi-frame　in　a　general　pair　of　dual　reducing　subspaces　of　(H-s　(R-d),　H-s　(R-d)).　We　characterize　reducing　subspaces　of　a　Sobolev　space.　Then,　under　the　setting　of　reducing　subspaces,　we　study　the　properties　of　weak　nonhomogeneous　wavelet　bi　frames,　obtain　a　construction　of　weak　nonhomogeneous　wavelet　bi-frames　from　a　pair　of　refinable　functions,　and　derive　a　fast　wavelet　algorithm　associated　with　such　weak　nonhomogeneous　wavelet　bi-frames.