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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c4">l</font>
 ;<br/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> )
 ;<br/>

<b>assume </b><a NAME="E3:9"/>
<font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font>
 ;<br/>

<a NAME="E4:9"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="nat_4.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>


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