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<b>thus </b><a NAME="E1:13_1_1"/>
not  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">u</font> =  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a><br/>

<a NAME="E1:13_1_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T27">POLYNOM1:27</a></i>;<br/>
<a NAME="E2:13_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E3:13_1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K27">1_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>;<br/>
<a NAME="E4:13_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:13_1_1"/>
 <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> is <a href="vectsp_1.html#V7">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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">n</font>,<font color="Maroon">L</font></span>)</span>, <font color="Maroon">y</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">n</font>,<font color="Maroon">L</font></span>)</span>, <font color="Maroon">z</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">n</font>,<font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a><br/>

<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font>, <font color="Maroon">v</font> = <font color="Maroon">y</font>, <font color="Maroon">w</font> = <font color="Maroon">z</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</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">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<a NAME="E3:13_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E4:13_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">u</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:13_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">u</font> <a href="polynom1.html#K28">*'</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">v</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T85">POLYNOM1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>

<a NAME="E7:13_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E8:13_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">v</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:13_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">w</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">w</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">u</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>


</div><b>end;</b></div>


</div>
