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<span class="kw">set </span><font color="Maroon" title="c3">Pm</font> =  <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>);<br/>
<span class="kw">thus </span><a NAME="E1:64_1_1"/>
 <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>) is  <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>) is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<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></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:64_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<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></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="polynom1.html#E41"><span class="lab"><font color="Green" title="E33">Lm1</font></span></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">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D2" title="VECTSP_1:def.2">VECTSP_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">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="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c6">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#D10" target="_self" title="POLYNOM1:def.10">Def27</a></span>;<br/>
<a NAME="E3:64_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c9">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def27</a></span>;<br/>
<a NAME="E4:64_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def27</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c7">p</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c9">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c9">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T26" target="_self" title="POLYNOM1:th.26">Th85</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">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="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def27</a></span>
;<br/> <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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