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<b>set </b><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="E1:58_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#K29" title="POLYNOM1:func.29">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="group_1.html#K9" title="GROUP_1:func.9">*</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="group_1.html#K9" title="GROUP_1:func.9">*</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="58_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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#K29" title="POLYNOM1:func.29">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:()"><i><font color="Red">::  thesis: </font></i></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="group_1.html#K9" title="GROUP_1:func.9">*</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="group_1.html#K9" title="GROUP_1:func.9">*</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/>

<b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c6">q</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> ;<br/>
<i><font color="Green" title="E41">A1</font></i>: <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="group_1.html#K9" title="GROUP_1:func.9">*</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">p'</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c8">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c8">q'</font></span>)</span>,<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="polynom2.html#D5" target="_self" title="POLYNOM2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom2.html#T27" target="_self" title="POLYNOM2:th.27">Th27</a></i>
;<br/>
<a NAME="E4:58_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> <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">p'</font>,<font color="Maroon" title="c3">x</font> &amp; <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="c10">q</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="c8">q'</font>,<font color="Maroon" title="c3">x</font> )
 
<b>by </b><i><a class="ref" href="polynom2.html#D5" target="_self" title="POLYNOM2:def.5">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" 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="group_1.html#K9" title="GROUP_1:func.9">*</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="group_1.html#K9" title="GROUP_1:func.9">*</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>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:58_1_1_1"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:58_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="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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