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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="gcd_1.html#V2" target="_self">multiplicative</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E2:45_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45"><i><font color="Green">E35</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:45_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T22" target="_self">Th22</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:45_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="gcd_1.html#V1" target="_self">unital</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E4:45_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="gcd_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E7:45_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green">E39</font></i></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">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="gcd_1.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E9:45_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E2:45_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:45_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:45_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E5:45_1_1_1_1"><i><font color="Green">E44</font></i></a>, <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">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</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">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E7:45_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gcd_1.html#E8:45_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="gcd_1.html#E9:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self">Th22</a></i>
;<br/><a NAME="E11:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:45_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="gcd_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E12:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:45_1_1_1"><i><font color="Green">E39</font></i></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">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E2:45_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45"><i><font color="Green">E35</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:45_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:45"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gcd_1.html#E1:45_1_1_2"><i><font color="Green">E36</font></i></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">c<sub>1</sub></font> is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:45_1_1_2"><i><font color="Green">E37</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">c<sub>3</sub></font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
