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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1" target="_self">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">s</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font></span>)</span>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> (  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp;  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">t</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> )
 ;<br/>

<b>thus </b><a NAME="E3:10"/>
for <font color="Olive">r</font>, <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">s</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" target="_self">GROUP_1:def 4</a><br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:10"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:10"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s2</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:10"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:10"/><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">t2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">t2</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E57</font></i>: <font color="Maroon">s1</font> = 
<font color="Maroon">s2</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">t1</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">s2</font>
<b>by </b><i>, , , </i>
;<br/>
<b>take </b>
<font color="Maroon">s1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" target="_self">GROUP_1:def 3</a><br/>

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

<a NAME="E13:10"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">t</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:10"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E15:10"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">r</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:10"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">s1</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/>

<b>consider </b><font color="Maroon">t1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E18:10"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E20:10"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">t2</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E22:10"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 <b>by </b><i/>;<br/>
<a NAME="E23:10"/><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">r2</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
<b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<font color="Maroon">t1</font> = 
<font color="Maroon">s1</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">t2</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">t1</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">t2</font>
<b>by </b><i>, , , </i>
;<br/>
<b>hence </b><a NAME="E25:10"/>
( <font color="Maroon">s</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">t1</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, </i>;<br/>


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