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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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>assume </b><a NAME="E1:45"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">Amp</font> is <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font>;<br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</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><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" title="c3">x</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="c3">x</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="45_1_1_1"><b>case </b></a><a NAME="E1:45_1_1_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">x</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="c5">d</font> = <font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font>;<br/><a NAME="E2:45_1_1_1"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">d'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font><b> such that </b><br/><a NAME="E4:45_1_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c6">d'</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">u</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="E6:45_1_1_1"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c7">u</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d'</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E4:45_1_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="gcd_1.html#T18" target="_self" title="GCD_1:th.18">Th18</a></i>;<br/><a NAME="E7:45_1_1_1"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">d'</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green" title="E39">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="E8:45_1_1_1"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:45"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="gcd_1.html#D10" target="_self" title="GCD_1:def.10">Def10</a></i>;<br/><a NAME="E9:45_1_1_1"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">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" title="c3">x</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font>
 
;<br/>
<a NAME="E2:45_1_1_1_1"/><b>then </b><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></i>;<br/>
<a NAME="E3:45_1_1_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="gcd_1.html#D2" target="_self" title="GCD_1:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e</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="E5:45_1_1_1_1"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c8">e</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>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c8">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">e</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="gcd_1.html#E5:45_1_1_1_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
;<br/>
<a NAME="E7:45_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</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="E8:45_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45_1_1_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div><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> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font> = 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c7">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="gcd_1.html#E7:45_1_1_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="gcd_1.html#E8:45_1_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gcd_1.html#E9:45_1_1_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>
;<br/><a NAME="E11:45_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">u</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>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:45_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="gcd_1.html#T1" target="_self" title="GCD_1:th.1">Th1</a></i>;<br/><a NAME="E12:45_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <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="E13:45_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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" title="E36">A12</font></i>: 
<font color="Maroon" title="c3">x</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"><b>set </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font>;<br/><a NAME="E2:45_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><a NAME="E3:45_1_1_2"/><b>then </b><i><font color="Green" title="E37">A13</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</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#E2:45"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gcd_1.html#E1:45_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/><a NAME="E4:45_1_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</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#T24" target="_self" title="GCD_1:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:45_1_1_2"><i><font color="Green" title="E37">A13</font></i></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" title="c3">x</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 ;<br/>


</div>
