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<b>let </b><font color="Maroon">R</font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">Amp</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green">E20</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:64"/><i><font color="Green">E21</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:64"/>
<font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E4:64"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="ref" href="gcd_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:64"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:64"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">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">E26</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T4" target="_self">Th4</a></i>;<br/>
<font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font></span>)</span> = 
<font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_1"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, <a class="ref" href="vectsp_1.html#D13">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" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E9:64"/><i><font color="Green">E54</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E10:64"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:64"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, <a class="ref" href="gcd_1.html#T10" target="_self">Th10</a>, </i>;<br/>


</div>
