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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="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">r</font> = <font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span>;<br/>
<a NAME="E1:23"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="pre_circ.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="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E2:23"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="pre_circ.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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/>
<b>hence </b><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span> = 
<font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
<font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>
;<br/><br/>

<b>set </b><font color="Maroon">e</font> = <font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:23"/><i><font color="Green">E40</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#K12">Support</a> <font color="Maroon">s</font></span>)</span> <a href="pre_circ.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> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <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>
<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="23_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_1_1"/>
( <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font> or  not <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1"><b>case </b></a><a NAME="E1:23_3_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span> <a href="pre_circ.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> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:23_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="polynom1.html#K15">.</a> <font color="Maroon">u'</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_2"><b>case </b></a><a NAME="E1:23_3_1_1_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E3:23_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> <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#D9">POLYNOM1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:23_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="polynom1.html#K15">.</a> <font color="Maroon">u'</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon">s</font> <a href="groeb_3.html#K2" target="_self">|</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</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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