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<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>thus </b><a NAME="E1:11"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_1_1"/>
<font color="Maroon">b</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_1_1"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_1"><i><font color="Green">E11</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_1"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_1_2"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_1"><i><font color="Green">E11</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:11"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:11"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> implies  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_2"/>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_2_2"/><i><font color="Green">E23</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_2_2"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_2_2"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_2"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:11"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<a NAME="E1:11_3"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> implies  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_3_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_3_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_3"/>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_2"/><i><font color="Green">E28</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:11_3_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_3_2"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11"><i><font color="Green">E10</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_3"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> iff  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:11"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b</font> iff  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="txt" href="int_2.html#E1:11"><i><font color="Green">E10</font></i></a></i>;<br/>


</div>
