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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>3</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="binop_1.html#V1">commutative</a> )
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:38"><i><font color="Green">E18</font></i></a>, <a class="txt" href="matrix_3.html#E2:38"><i><font color="Green">E19</font></i></a>, <a class="ref" href="matrix_3.html#T25" target="_self">Th25</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:38"><i><font color="Green">E18</font></i></a>, <a class="txt" href="matrix_3.html#E2:38"><i><font color="Green">E19</font></i></a>, <a class="ref" href="matrix_3.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funcop_1.html#K8">[;]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E2:38"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funcop_1.html#T79">FUNCOP_1:79</a></i>
;<br/>


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