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<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</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="E3:44"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:44"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:44_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E3:44_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:44_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:44_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:44"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E6:44_1"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:44_1"/><b>then </b>
<font color="Maroon">h</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E1:44"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E3:44"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="nat_1.html#E5:44"><i><font color="Green">E34</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:44_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:44"><i><font color="Green">E32</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:44"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self">Th2</a>, </i>;<br/>


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