## Appendix D
In addition, for \"\(\mathbf{M_{0}}\)\", there is an actor named \"a\" and \"\(\mathbf{M_{0}}\)\" is divided into four (4) sub-missions \"\(\mathbf{M_{(t,a)}}\mathbf{:W^{(j)}}\)\", where \(\mathbf{t\in[1,4]}\). In this case, according to Equations (A2) and (A3), the process of thematic decomposition could be described as the following:

\[\mathbf{M_{(0,1)}}=\mathbf{M_{(1,1)}}+\mathbf{M_{(2,1)}}+\mathbf{M_{(2,1)}}+ \mathbf{M_{(3,1)}}+\mathbf{M_{(4,1)}}\] (A4)

In the case of having \"x\" sub-missions (thematic decomposition) and 1 actor, Equation (A3) could be generalized as follows:

\[\mathbf{M_{(0,1)}}=\mathbf{M_{(1,1)}}+\mathbf{M_{(2,1)}}+\mathbf{M_{(3,1)}}+ \mathbf{M_{(4,1)}}+\ldots+\mathbf{M_{(x,1)}}\ \mathrm{or}\mathbf{M_{(0,1)}}=\sum_{i=0}^{x} \mathbf{M(x,1)},\mathrm{where}\ \mathrm{x}\in\mathbb{N}^{*}\] (A5)

The summation represented in Equation (A5) can also be implemented in programming languages such as C, C++, JavaScript, or Python. Below is an example of the implementation in Python:

_E4. A Mission, with Participant p-Actors and Multi-Thematic Decomposition into Sub-Missions_

Often, the great majority of missions are extremely complicated, highly diverse, highly polymorphic, multi-thematic, and include several actors. Consequently, each p-actor must use theme deconstruction by examining the original mission \"\(\mathbf{M_{(0,p)}}\): \(\mathbf{W^{(j)}}\)\" and dividing it into further sub-missions until \"\(\mathbf{M_{(0,p)}}\): \(\mathbf{W^{(j)}}\)\" cannot be analyzed anymore because the last elementary sub-mission has been reached. Suppose that:

* p-actors, p \(\in[1,2,3,\ldots,\ n]\), where n is the max number of actors, n \(\in\mathbb{N}^{*}\),
* x is the max number of sub-missions of 1 actor, where x \(\in\mathbb{N}^{*}\),
* k is the max number of sub-missions of 3 actors, where k \(\in\mathbb{N}^{*}\),
* z is the max number of sub-missions of \"n\" actors, where z \(\in\mathbb{N}^{*}\),
* M\({}_{(0,p)}\) is the initial mission \"M\({}^{0}\)\" of each participant p-actor, where p \(\in\) [a,n]. Applying Equation (A2), it is proven that:

\[\mathbf{M_{(0,a)}}=\mathbf{M_{(1,1)}}+\mathbf{M_{(2,1)}}+\mathbf{M_{(3,1)}}+ \mathbf{M_{(4,1)}}+\ldots+\mathbf{M_{(c,1)}}\] (A6)

\[\mathbf{M_{(0,b)}}=\mathbf{M_{(1,2)}}+\mathbf{M_{(2,2)}}+\mathbf{M_{(3,2)}}+ \mathbf{M_{(4,2)}}+\ldots+\mathbf{M_{(c,2)}}\] (A7)

\[\mathbf{M_{(0,c)}}=\mathbf{M_{(1,3)}}+\mathbf{M_{(2,3)}}+\mathbf{M_{(3,3)}}+ \mathbf{M_{(4,3)}}+\ldots+\mathbf{M_{(c,3)}}\] (A8)

\[\mathbf{M_{(0,n)}}=\mathbf{M_{(1,n)}}+\mathbf{M_{(2,n)}}+\mathbf{M_{(3,n)}}+ \mathbf{M_{(4,n)}}+\ldots+\mathbf{M_{(c,n)}}\] (A9)

Combining Equations (A2) and (A3) and applying them to Equations (A6)-(A9), for the whole mission it is proven that:

\[\sum(\mathbf{M})=\sum_{i=0}^{n}\mathbf{M(i,p)}=\sum_{i=0}^{x}\mathbf{M(i,1)}+ \sum_{i=0}^{r}\mathbf{M(i,2)}+\sum_{i=0}^{k}\mathbf{M(i,3)}+\ldots+\sum_{i=0 }^{z}\mathbf{M(i,n)}\] (A10)

The summation represented in Equation (A10) can also be implemented in programming languages such as C, C++, JavaScript, or Python. Below is an example of the implementation in Python:

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