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<b>let </b><font color="Maroon">R</font> be  <a href="group_1.html#V7">commutative</a> <a href="vectsp_2.html#V2">domRing-like</a> <a href="funcsdom.html#NM1">Ring</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>assume </b><a NAME="E1:45"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">Amp</font> is <a href="gcd_1.html#V2" target="_self">multiplicative</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>;<br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">y</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>case </b></a><a NAME="E1:45_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">d</font> = <font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font>;<br/><a NAME="E2:45_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">d'</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font><b> such that </b><br/><a NAME="E4:45_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">d'</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:45_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">u</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#D4" target="_self">Def4</a>, <a class="txt" href="gcd_1.html#E1:45"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E7:45_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d'</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self">Def3</a>, <a class="ref" href="gcd_1.html#T2" target="_self">Th2</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E8:45_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E9:45_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_1_1"/>
<font color="Maroon">x</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E2:45_1_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:45_1_1_1_1"/>
<font color="Maroon">u</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>then 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="E5:45_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<a NAME="E7:45_1_1_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:45_1_1_1_1"/>
<font color="Maroon">x</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> = 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
<b>by </b><i>, , , </i>
;<br/><a NAME="E11:45_1_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E12:45_1_1_1"/><b>then </b>
<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E13:45_1_1_1"/>
<font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><b>case </b></a><a NAME="E1:45_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">d</font> = <font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font>;<br/><a NAME="E2:45_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:45_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><a NAME="E4:45_1_1_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:45_1_1_2"/>
<font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon">x</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 ;<br/>


</div>
