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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T8" target="_self" title="NAT_1:th.8">Th8</a></i>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E10">A1</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="c2">j</font>
 ;<br/>

<b>hence </b><a NAME="E3:25"/>
<font color="Maroon" title="c1">i</font> <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#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" target="_self" title="NAT_1:th.12">Th12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a><br/>

<b>assume </b><a NAME="E4:25"/><i><font color="Green" title="E11">A2</font></i>: 
<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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/>

<a NAME="E5:25"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E6:25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<b>hence </b><a NAME="E7:25"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E2:25"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="nat_1.html#E4:25"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nat_1.html#E5:25"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>


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