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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="gcd_1.html#NM3" title="GCD_1:NM.3">gcdDomain</a>;<br/><b>let </b><font color="Maroon" title="c2">Amp</font> be    <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">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/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1"/>
( <font color="Maroon" title="c5">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>case </b></a><a NAME="E1:63_1_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c6">D</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1_1_1_1"/>
(  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:63_1_1_1_1_1_1"/><i><font color="Green" title="E47">A2</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c7">E</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>;<br/><a NAME="E2:63_1_1_1_1_1_1"/>
(  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E3:63_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#T5" target="_self" title="GCD_1:th.5">Th5</a></i>;<br/><a NAME="E4:63_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">F</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="E6:63_1_1_1_1_1_1"/><i><font color="Green" title="E48">A3</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/><a NAME="E7:63_1_1_1_1_1_1"/><i><font color="Green" title="E49">A4</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E8:63_1_1_1_1_1_1"/><i><font color="Green" title="E50">A5</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E9:63_1_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c9">G</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="E2:63_1_1_1_1_1_1_1"/><i><font color="Green" title="E51">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:63_1_1_1_1_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="gcd_1.html#E7:63_1_1_1_1_1_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>
<a NAME="E3:63_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:63_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E4:63_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">A</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="c10">G</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="E6:63_1_1_1_1_1_1_1"/><i><font color="Green" title="E53">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:63_1_1_1_1_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="gcd_1.html#E8:63_1_1_1_1_1_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>
<a NAME="E7:63_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E6:63_1_1_1_1_1_1_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E8:63_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">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>hence </b><a NAME="E9:63_1_1_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:63_1_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="gcd_1.html#E4:63_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="gcd_1.html#T15" target="_self" title="GCD_1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:63_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E11:63_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:63_1_1_1_1_1_1_2"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:63_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">F</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:63_1_1_1_1_1_1"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="gcd_1.html#E10:63_1_1_1_1_1_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="gcd_1.html#T15" target="_self" title="GCD_1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:63_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">F</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self" title="GCD_1:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E13:63_1_1_1_1_1_1"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:63_1_1_1_1_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="gcd_1.html#T18" target="_self" title="GCD_1:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:63_1_1_1_1_1_2"/><i><font color="Green" title="E47">A10</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:63_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></i>;<br/><a NAME="E3:63_1_1_1_1_1_2"/><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:63_1_1_1_1_1_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/> <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> = 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<b>by </b><i><a class="txt" href="gcd_1.html#E2:63_1_1_1_1_1_2"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E3:63_1_1_1_1_1_2"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="gcd_1.html#T32" target="_self" title="GCD_1:th.32">Th32</a></i>
;<br/><b>hence </b><a NAME="E5:63_1_1_1_1_1_2"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:63_1_1_1"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>case </b></a><a NAME="E1:63_1_1_2"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:63_1_1_2"/><b>then </b><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><a NAME="E3:63_1_1_2"/>
<font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:63_1_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><b>then </b> <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="gcd_1.html#E2:63_1_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="gcd_1.html#T32" target="_self" title="GCD_1:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:63_1_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
;<br/><b>hence </b><a NAME="E5:63_1_1_2"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:63"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 ;<br/>


</div>
