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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/>

<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon">L</font> is   <a href="algstr_1.html#NM4">multGroup</a> implies ( ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> ) &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">c</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="algstr_1.html#T3" target="_self">Th3</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E24</font></i>: 
( ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> ) &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">c</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:38_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="algstr_1.html#E13"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:38_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:38_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</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> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">x</font> = <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="algstr_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:38_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:38_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</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">L</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="algstr_1.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">x</font> = <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="algstr_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:38_1"/>
for <font color="Olive">a</font>, <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:38_1_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</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">L</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="algstr_1.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:38_1_4"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
;<br/>
<b>hence </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="algstr_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:38_1"/>
for <font color="Olive">a</font>, <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:38_1_5"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:38_1_5"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 ;<br/>

<b>then </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
;<br/>
<b>hence </b><font color="Maroon">x</font> = 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="algstr_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon">L</font> is   <a href="algstr_1.html#NM4">multGroup</a>
 <b>by </b><i><a class="txt" href="algstr_1.html#E16"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="algstr_1.html#T3" target="_self">Th3</a>, <a class="ref" href="algstr_1.html#T5" target="_self">Th5</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D2">GROUP_1:def 2</a></i>;<br/>


</div>
