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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E38">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E39">A2</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font></span><br/>

<b>assume </b><a NAME="E3:82"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:82"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 
<b>by </b><i><a class="txt" href="int_1.html#E1:82"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/>
<a NAME="E5:82"/>
<font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D4" target="_self" title="INT_1:def.4">Def4</a></i>;<br/>
<a NAME="E6:82"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E2:82"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:82"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E2:82"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E8:82"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E3:82"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E4:82"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
;<br/>
<a NAME="E11:82"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:82"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="int_1.html#E3:82"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:82"/>
contradiction
 <b>by </b><i><a class="txt" href="int_1.html#E1:82"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="int_1.html#E2:82"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="int_1.html#E3:82"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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