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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>set </b><font color="Maroon">D</font> =  <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>;<br/>
<a NAME="E1:66"/><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">C</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font> &amp;  <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#R2" target="_self">divides</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:66"/><i><font color="Green">E21</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> 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>consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:66"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><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></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:66"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><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></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E7:66"/><i><font color="Green">E53</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#R2" target="_self">divides</a> <font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><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></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="group_1.html#K10">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">E</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="group_1.html#K10">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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>
<b>by </b><i><a class="ref" href="gcd_1.html#T10" target="_self">Th10</a>, <a class="ref" href="gcd_1.html#T11" target="_self">Th11</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_1"/>
 <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#R2" target="_self">divides</a> <font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:66"/>
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">C</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">Z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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> &amp; <font color="Maroon">Z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">C</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:66_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:66_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:66_2"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">Y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">Y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="group_1.html#K10">*</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T40">VECTSP_1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><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> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E7:66_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:66_2"/>
<font color="Maroon">Z</font> <a href="gcd_1.html#R2" target="_self">divides</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="txt" href="gcd_1.html#E1:66_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:66"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <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></span>)</span>,<font color="Maroon">C</font>,<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#T8" target="_self">Th8</a>, <a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, <a class="txt" href="gcd_1.html#E7:66"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>


</div>
