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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>









<b>assume </b><a NAME="E1:66"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1"/>
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>suppose </b></a><a NAME="E1:66_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_1.html#E1:66"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="group_1.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><b>suppose </b></a><a NAME="E1:66_1_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a>  )
 <b>by </b><i><a class="txt" href="group_1.html#E1:66"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:66_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="group_1.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/><b>hence </b><a NAME="E4:66_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3"><b>suppose </b></a><a NAME="E1:66_1_3"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_1_3"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a>  &amp; <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_1.html#E1:66"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:66_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="group_1.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/><b>hence </b><a NAME="E4:66_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4"><b>suppose </b></a><a NAME="E1:66_1_4"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_1_4"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a>  &amp; <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a>  )
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E2:66_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_1.html#E2"><i><font color="Green">Lemma38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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