This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors: (1) Tomáš Jakl, Czech Academy of Sciences and Czech Technical University; (2) Dan Marsden, School of Computer Science University of Nottingham; (3) Nihil Shah, Department of Computer Science University of Oxford. Table of Links Abstract & Introduction Prelimenaries FVM Theorems for Positive Existential Fragments FVM Theorems for Counting Logic FVM Theorems for The Full Logic Abstract FVM Theorems for Products Adding Equality and Other Enrichment Conclusions, Acknowledgments & References Appendix A FVM theorems for coproducts Appendix B Proofs Omitted from Section III Appendix C Proofs Omitted from Section IV Appendix D Proofs Omitted from Section V Appendix E Proofs Omitted from Section VI Appendix F Proofs Omitted from Section VII APPENDIX D PROOFS OMITTED FROM SECTION V

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