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<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<b>thus </b><a NAME="E1:11"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">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" title="E10">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">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" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_1"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">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" title="E11">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:11"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">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" title="E11">A6</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> implies  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">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" title="E12">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_2_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">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" title="E12">A8</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_2_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_2_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_2"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_2"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:11"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">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" title="E11">A9</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> implies  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">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" title="E12">A10</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_3_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:11"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="int_2.html#E1:11_3_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_3"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">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" title="E12">A11</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:11_3_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T11" target="_self" title="INT_2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E2:11"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="int_2.html#E1:11_3_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self" title="INT_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_3"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> iff  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:11_3"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E4:11"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> iff  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:11"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="int_2.html#E2:11"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>


</div>
