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<div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM3">Skew-Field</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">R</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>set </b><font color="Maroon">ct</font> =  <a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font>;<br/>
<b>set </b><font color="Maroon">cs</font> =  <a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">cct</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">ccs</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span>;<br/>
<a NAME="E2:42"/><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="uniroots.html#D1">UNIROOTS:def 1</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">R</font></span>)</span> : <font color="Maroon">t</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">t</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:42"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:42_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_2"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="txt" href="weddwitt.html#E3:42"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><font color="Maroon">t</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c1</font> = 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:42"><i><font color="Green">E36</font></i></a>, <a class="ref" href="uniroots.html#T19">UNIROOTS:19</a></i>
<br/>.= 
<font color="Maroon">c1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">t</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:42"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="uniroots.html#T19">UNIROOTS:19</a></i>
;<br/><a NAME="E5:42_1_1_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:42"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E6:42_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:42_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1"><b>suppose </b></a><a NAME="E1:42_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E3:42_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:42_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">t</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:42"><i><font color="Green">E38</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="uniroots.html#T22">UNIROOTS:22</a></i>;<br/><font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
<font color="Maroon">t</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:42"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="uniroots.html#T19">UNIROOTS:19</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:42"><i><font color="Green">E36</font></i></a>, <a class="ref" href="uniroots.html#T19">UNIROOTS:19</a></i>
;<br/><b>hence </b><a NAME="E7:42_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_2"><b>suppose </b></a><a NAME="E1:42_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:42_1_2_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
