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

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">l</font>
 ;<br/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> )
 ;<br/>

<b>assume </b><a NAME="E3:9"/>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E4:9"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
<font color="Maroon">j</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">l</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
contradiction
 <b>by </b><i/>;<br/>


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