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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>thus </b><a NAME="E2:124"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> implies <font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:124_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:124_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:124_1"/>
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:124_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:124_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_2_1_1"><b>suppose </b></a><a NAME="E1:124_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:124_2_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_2_1_2"><b>suppose </b></a><a NAME="E1:124_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:124_2_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:124_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">nn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><a NAME="E5:124_2_1_2"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T136">XREAL_1:136</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:124_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:124_2_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


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