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<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E1:58_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
 <a href="polynom2.html#K5" target="_self">eval</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom2.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom2.html#T25" target="_self">Th25</a></i>
;<br/>
<a NAME="E4:58_1_1_1"/>
( <span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:58_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:58_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:58_1_1"/>
 <a href="polynom2.html#K6" target="_self">Polynom-Evaluation</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="grcat_1.html#V1">additive</a>
 <b>by </b><i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>


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