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<div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_2.html#NM3">Skew-Field</a>;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font>;<br/>
<a NAME="E1:17"/>
for <font color="Olive">x</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> st <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> ) &amp;  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span>;<br/>

<a NAME="E1:17_1"/><i><font color="Green">E26</font></i>: 
(  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:17_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> st <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:17_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_2.html#T40">VECTSP_2:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:17_1_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> st <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> ) &amp;  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
 <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font> is   <a href="vectsp_2.html#NM3">Skew-Field</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>


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