Table 6. Theorem representing the feed forward loop: A & A & B & C & RA& RA& RB ⊢ C.
| 1. A & A & B & C & RA & RA & RB | IA |
| 2. A | From 1 by &E |
| 3. A | From 1 by &E |
| 4. B | From 1 by &E |
| 5. C | From 1 by &E |
| 6. RA | From 1 by &E |
| 7. RA | From 1 by &E |
| 8. RB | From 1 by &E |
| 9. A &RA | From 2,6 by →I |
| 10. A & RA → A*RA | EVF |
| 11. A*RA | From 9,10 by →E |
| 12. A*RA → A | EVF |
| 13. A | From 11,12 by → E |
| 14. A & RB | From 8,13 by &I |
| 15. A & RB → A*RB | EVF |
| 16. A*RB | From 14,15 by →E |
| 17. (A*RB) & B | From 4,16 by &I |
| 18. (A*RB) & B → (A*RB) *B | EVF |
| 19. (A*RB) *B | From 17,18 by →E |
| 20. (A*RB) *B → B | EVF |
| 21. B | From 19,20 by →E |
| 22. A & RA | From 3,7 by &I |
| 23. A & RA → A*RA | EVF |
| 24. A*RA | From 22,23 by →E |
| 25. A*RA → A | EVF |
| 26. A | From 24,25 by →E |
| 27. A & B | From 21,26 by &I |
| 28. A & B → A*B | EVF |
| 29. A*B | From 27,28 by →E |
| 30. (A*B) & C | From 5,29 by &I |
| 31. (A*B) & C → (A*B) *C | EVF |
| 32. (A * B) *C | From 30,31 by →E |
| 33. (A * B) *C → C | EVF |
| 34. C | From 32,33 by →E |
| 35. ( A & A & B & C & RA & RA & RB) → C | From 1–34 by →I |
A feed forward loop [30] is illustrated in the detailed form. In this form, the theorem is reported on line 35, the antecedent (IA, initial aggregate) is the multiset reported on line 1 and is “discharged” by the application of →I. The theorem illustrates the abstract case of a feed forward loop composed of three genes A, B, C, their encoded proteins (A, B, C), and two regulatory proteins RA and RB, such that (i) A is regulated by RA; (ii) B by RB and the protein A; (iii) C by the protein complex A*B. The reader can check that each formula (resource) is used at most once.