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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">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">a</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Olive" title="b3">c</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c3">B</font> &amp; <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="c4">C</font> implies <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="c4">C</font> )</span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E9">A1</font></i>: 
( <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="c3">B</font> &amp; <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="c4">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c3">B</font> &amp; <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="c4">C</font> implies <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="c4">C</font> )</span><br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/><a NAME="E3:16_1"/>
( <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/><a NAME="E4:16_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="#E2:16_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E5:16_1"/>
( <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="c3">B</font> &amp; <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="c4">C</font> implies <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="c4">C</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
( <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="c3">B</font> &amp; <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="c4">C</font> implies <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="c4">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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