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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:34"/>
<span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E2:34"/><b>then </b>
 <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:34"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">Z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="gcd_1.html#E4:34"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E6:34"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E7:34"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E8:34"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b>
<font color="Maroon" title="c5">Z</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E6:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E7:34"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:34"/>
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:34_2"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E3:34_2"/><b>then </b>
<font color="Maroon" title="c5">Z</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E7:34"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/>
<a NAME="E4:34_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E6:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:34_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
 <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:34"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>;<br/>


</div>
