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

<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#R1" title="GCD_1:pred.1">divides</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</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#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="c4">C</font> implies <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> )</span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E7">A1</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="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#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="c4">C</font> implies <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> )</span><br/><b>then consider </b><font color="Maroon" title="c5">D</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">E</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="#E1:14_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/><a NAME="E6:14_1"/>
<font color="Maroon" title="c2">A</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E5:14_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:14_1"/>
( <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="c4">C</font> implies <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> )
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</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="E2:14"/>
( <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="c4">C</font> implies <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
