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<div class="add">

<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/>

<a NAME="E1:30"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">a</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">b</font> holds <br/> ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Olive">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<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>;<br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">A</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">A</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1"/>
( <font color="Maroon">A</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">A</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="30_1_1_1_1"><b>case </b></a><a NAME="E1:30_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</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"><a NAME="E2:30_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E3:30_1_1_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</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="txt" href="gcd_1.html#E4:30_1"><i><font color="Green">E25</font></i></a>, </i>;<br/><a NAME="E4:30_1_1_1_1"/><b>then </b>
<font color="Maroon">C</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/>;<br/><a NAME="E5:30_1_1_1_1"/><b>then </b>
<font color="Maroon">C</font> is <a href="gcd_1.html#V1" target="_self">unital</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:30_1_1_1_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><b>case </b></a><a NAME="E1:30_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_2"/><b>then </b>
<font color="Maroon">B</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="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/><a NAME="E3:30_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <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><a class="txt" href="gcd_1.html#E4:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><a NAME="E4:30_1_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> is <a href="gcd_1.html#V1" target="_self">unital</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1_1_1_2_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="gcd_1.html#D2" target="_self">GCD_1:def 2</a><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:30_1_1_1_2"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:30_1"><i><font color="Green">E26</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="E8:30_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Olive">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) holds <br/><font color="Olive">a</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<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>;<br/>

<a NAME="E1:30_2"/>
(  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) implies <font color="Maroon">A</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_1_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:30_2_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">C</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E4:30_2_1_1"/>
<font color="Maroon">C</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>, </i>;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:30_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</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/><font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="group_1.html#K10">*</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E8:30_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">B</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E9:30_2_1_1"/>
<font color="Maroon">A</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:30_2_1_1"/>
(  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) implies <font color="Maroon">A</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_2_1"/>
(  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) implies <font color="Maroon">A</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_2"/>
(  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Maroon">A</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) implies <font color="Maroon">A</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">a</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">b</font> iff  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <font color="Olive">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>


</div>
