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



<a NAME="E1:8"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#R1">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="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</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="E4:8_1"/>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</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>
<a NAME="E2:8"/>
( <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="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:8_2_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E3:8_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><b>hence </b><a NAME="E4:8_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="E6:8_2"/>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:8_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> iff <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div>
