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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    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:35"/><i><font color="Green">E25</font></i>: 
(  <a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp;  <a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp;  <a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a></i>;<br/>
<b>thus </b> <a href="rfunct_3.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finsop_1.html#T6">FINSOP_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" target="_self">addpfunc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rfunct_3.html#K6" target="_self">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a></i>
;<br/>


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