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<b>thus </b><a NAME="E1:13_1_1"/>
not  <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> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">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" title="c3">u</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_1.html#D8" title="ANPROJ_1:def.8">ANPROJ_1:def 8</a><br/>

<a NAME="E1:13_1_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T27" title="POLYNOM1:th.27">POLYNOM1:27</a></i>;<br/>
<a NAME="E2:13_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <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/>
<a NAME="E3:13_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>;<br/>
<a NAME="E4:13_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:13_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="polynom6.html#E3:13_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>;<br/>


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

<b>thus </b><a NAME="E2:13_1_1"/>
 <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> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">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" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">w</font> = <font color="Maroon" title="c5">z</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">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c11">z'</font> = <font color="Maroon" title="c5">z</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/>
<a NAME="E3:13_1_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c10">y'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c8">w</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E4:13_1_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:13_1_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">z'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c6">u</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c8">w</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:13_1_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c8">w</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E4:13_1_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="polynom6.html#E5:13_1_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/>

<a NAME="E7:13_1_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c10">y'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c8">w</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E8:13_1_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:13_1_1_2"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c8">w</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c6">u</font>
<b>by </b><i><a class="txt" href="polynom6.html#E7:13_1_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c6">u</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom6.html#T8" target="_self" title="POLYNOM6:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <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"><font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E8:13_1_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="polynom6.html#E9:13_1_1_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/>


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


</div>
