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<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="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6"/><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">t</font></span>)</span>
 <b>and </b><br/><a NAME="E2:6"/><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="E3:6"/>
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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_1.html#E1:6"><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="E6:6"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="E8:6"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c4">s2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="E10:6"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<a NAME="E11:6"/><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">t1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E8:6"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="group_1.html#E10:6"><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#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E6:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="group_1.html#E10:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="group_1.html#E11:6"><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="E13:6"/>
 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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:6"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E6:6"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>

<a NAME="E15:6"/>
 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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:6"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E8:6"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="group_1.html#E12:6"><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="E18:6"/><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="E20:6"/><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c8">t2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="E22:6"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<a NAME="E23:6"/><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E2:6"><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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E16:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="group_1.html#E22:6"><i><font color="Green" title="E16">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">t1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E20:6"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="group_1.html#E22:6"><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#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="group_1.html#E14:6"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="group_1.html#E18:6"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="group_1.html#E23:6"><i><font color="Green" title="E17">A13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E25:6"/>
( <font color="Maroon" title="c6">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E18:6"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="group_1.html#E20:6"><i><font color="Green" title="E15">A11</font></i></a></i>;<br/>


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