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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">t</font></span>)</span>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> (  ex <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Olive" title="b1">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> &amp;  ex <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Olive" title="b3">t</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> )
 ;<br/>

<b>thus </b><a NAME="E3:10"/>
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" target="_self" title="GROUP_1:def.4">GROUP_1:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c2">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:10"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E8:10"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c4">s2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">t1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E10:10"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<a NAME="E11:10"/><i><font color="Green" title="E10">A6</font></i>: 
 ex <font color="Olive" title="b1">t2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Olive" title="b1">t2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<i><font color="Green" title="E11">A7</font></i>: <font color="Maroon" title="c3">s1</font> = 
<font color="Maroon" title="c4">s2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">t1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E8:10"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="group_1.html#E10:10"><i><font color="Green" title="E9">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">s2</font>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E6:10"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="group_1.html#E10:10"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="group_1.html#E11:10"><i><font color="Green" title="E10">A6</font></i></a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c3">s1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" target="_self" title="GROUP_1:def.3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<a NAME="E13:10"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Olive" title="b1">t</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:10"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E6:10"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>

<a NAME="E15:10"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon" title="c2">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:10"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E8:10"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="group_1.html#E12:10"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon" title="c7">t1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E18:10"/><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">t2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E20:10"/><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c8">t2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E22:10"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<a NAME="E23:10"/><i><font color="Green" title="E17">A13</font></i>: 
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Olive" title="b1">r2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">t1</font>
;<br/>

<font color="Maroon" title="c7">t1</font> = 
<font color="Maroon" title="c3">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E16:10"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="group_1.html#E22:10"><i><font color="Green" title="E16">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">t1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E20:10"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="group_1.html#E22:10"><i><font color="Green" title="E16">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t2</font>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E14:10"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="group_1.html#E18:10"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="group_1.html#E23:10"><i><font color="Green" title="E17">A13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E25:10"/>
( <font color="Maroon" title="c6">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> )
 <b>by </b><i><a class="txt" href="group_1.html#E18:10"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="group_1.html#E20:10"><i><font color="Green" title="E15">A11</font></i></a></i>;<br/>


</div>
