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 A FVM THEOREMS FOR COPRODUCTS

In this section we give a detailed account of the FVM theorems for coproducts/disjoint union discussed in Examples III.4, IV.5, V.15. We explicitly prove FVM theorems for coproducts of arbitrary collection of structures with respect to logic equivalences captured by Pk. The argument for Ek is similar.