<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>

<a NAME="E1:15"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</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  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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>, <font color="Maroon">y</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>, <font color="Maroon">z</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/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font>, <font color="Maroon">c</font> = <font color="Maroon">z</font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> ;<br/>
<a NAME="E2:15_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E3:15_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
 <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">K</font> is   <a href="funcsdom.html#NM1">Ring</a>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>


</div>
