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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">i1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/>

<a NAME="E2:48"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">i0</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:48"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i0</font>
 ;<br/>
<a NAME="E5:48"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 
<b>by </b><i><a class="txt" href="int_1.html#E5:48"><i><font color="Green">E23</font></i></a></i>;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/>
( <font color="Maroon">i0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E2:48_1"/><b>then </b>
<font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:48_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_1.html#E4:48"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:48_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_1.html#E4:48"><i><font color="Green">E22</font></i></a>, <a class="txt" href="int_1.html#E5:48"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E2:48_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:48_2"/>
contradiction
 <b>by </b><i><a class="txt" href="int_1.html#E4:48"><i><font color="Green">E22</font></i></a>, <a class="txt" href="int_1.html#E5:48"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:48"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="int_1.html#E6:48"><i><font color="Green">E24</font></i></a>, </i>;<br/>


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