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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a>  <a href="vectsp_1.html#L2">multLoopStr_0</a> ;<br/>



<b>set </b><font color="Maroon">o1</font> = <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4" target="_self">|</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">o2</font> =  <a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<a NAME="E1:46"/><i><font color="Green">E27</font></i>: 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4" target="_self">|</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:46"/><i><font color="Green">E28</font></i>: 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">X</font>,<font color="Maroon">L</font> ;<br/><i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D24">POLYNOM1:def 24</a></i>
<br/>.= 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/><a NAME="E7:46_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>;<br/><a NAME="E8:46_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E9:46_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><i><font color="Green">E38</font></i>: <font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_3"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font> or <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3_1"><b>suppose </b></a><a NAME="E1:46_1_3_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4" target="_self">|</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3_2"><b>suppose </b></a><a NAME="E1:46_1_3_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:46_1_3_2"/><b>then </b>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <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/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>hence </b><a NAME="E4:46_1_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4" target="_self">|</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4" target="_self">|</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K27">1_</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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