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<span class="kw">thus </span><a NAME="E1:13_1_1"/>
not  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="struct_0.html#D19" title="STRUCT_0:def.19">STRUCT_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span></span><br/>

<a NAME="E1:13_1_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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#K10" title="POLYNOM1:func.10">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></span>;<br/>
<a NAME="E2:13_1_1_1"/>
(  <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> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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#K9" title="POLYNOM1:func.9">0_</a> <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#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1_1_1"/>
not <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:13_1_1"/>
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> )</span><br/>

<span class="kw">reconsider </span><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#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#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c10">y9</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c11">z9</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#K13" title="POLYNOM1:func.13">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"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( <font color="Maroon" title="c6">u</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y9</font> &amp; <font color="Maroon" title="c6">u</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">z9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E4:13_1_1_2"/>
<font color="Maroon" title="c10">y9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c8">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">hence </span><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#K11" title="POLYNOM1:func.11">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1_2"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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></span><br/>

<a NAME="E6:13_1_1_2"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
( <font color="Maroon" title="c7">v</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c8">w</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E7:13_1_1_2"/>
<font color="Maroon" title="c10">y9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c8">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="kw">hence </span><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#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c6">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c6">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom6.html#T8" target="_self" title="POLYNOM6:th.8">Th8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_1_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
