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<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  <a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>);<br/>
<a NAME="E1:52_1_1"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>)</span>)</span> holds  <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">p9</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c8">q9</font> = <font color="Maroon" title="c6">q</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c6">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>)</span>)</span> ;<br/>
<a NAME="E3:52_1_1_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D5" target="_self" title="POLYNOM2:def.5">Def5</a></span>;<br/>
<span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">q</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c8">q9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>
<br/>.= 
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c8">q9</font></span>)</span>,<font color="Maroon" title="c3">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D5" target="_self" title="POLYNOM2:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T23" target="_self" title="POLYNOM2:th.23">Th25</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:52_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="polynom2.html#D5" target="_self" title="POLYNOM2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:52_1_1"/>
 <a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">x</font>) is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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