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

<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/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E47">A1</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="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E2:64"/><i><font color="Green" title="E48">A2</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#K10" title="GROUP_1:func.10">*</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#K10" title="GROUP_1:func.10">*</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#K10" title="GROUP_1:func.10">*</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="ref" href="gcd_1.html#T37" target="_self" title="GCD_1:th.37">Th37</a></i>;<br/>
<a NAME="E3:64"/>
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:64"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E4:64"/><b>then </b>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:64"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="gcd_1.html#T35" target="_self" title="GCD_1:th.35">Th35</a></i>;<br/>
<a NAME="E5:64"/><b>then </b><i><font color="Green" title="E49">A3</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="c5">C</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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#T36" target="_self" title="GCD_1:th.36">Th36</a></i>;<br/>
<a NAME="E6:64"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1"/><i><font color="Green" title="E50">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#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T37" target="_self" title="GCD_1:th.37">Th37</a></i>;<br/>
<font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> = 
<font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font>
<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="E3:64_1"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="c3">A</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:64_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:64"/><b>then </b>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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#T35" target="_self" title="GCD_1:th.35">Th35</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green" title="E50">A5</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E5:64"><i><font color="Green" title="E49">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="E9:64"/><i><font color="Green" title="E51">A6</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <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="E10:64"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <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>hence </b><a NAME="E11:64"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:64"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="gcd_1.html#E9:64"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>;<br/>


</div>
