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<b>let </b><font color="Maroon" title="c1">j</font>, <font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon" title="c1">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">j</font></span>)</span> )
 <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" title="c1">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 
<b>by </b><i><a class="txt" href="nat_d.html#E3:8_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:8_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font>
 ;<br/><a NAME="E3:8_1_1"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:8_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="nat_d.html#T2" target="_self" title="NAT_D:th.2">Th47</a></i>;<br/><a NAME="E4:8_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">j</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:8_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="nat_d.html#E2:8_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th46</a></i>;<br/><b>hence </b><a NAME="E5:8_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E4:8_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="nat_d.html#E3:8_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="nat_1.html#T18" title="NAT_1:th.18">NAT_1:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:8_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E3:8_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">j</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E3:8"/>
<font color="Maroon" title="c1">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>


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