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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E22</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E3:82"/><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:82"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:82"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E6:82"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:82"/><b>then </b><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E8:82"/>
<font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E3:82"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E11:82"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:82"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E2"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:82"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
;<br/>
<a NAME="E14:82"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E15:82"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>


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