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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/><a NAME="E2:64_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T59" target="_self">Th59</a></i>;<br/><b>hence </b><a NAME="E3:64_1"/>
( <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_2"/>
<font color="Maroon">c<sub>1</sub></font> is  not   <a href="int_1.html#NM1">Integer</a>
 ;<br/><a NAME="E2:64_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E3:64_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:64_2"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="int_1.html#E26"><i><font color="Green">Lemma12</font></i></a></i>;<br/><a NAME="E5:64_2"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:64_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E7:64_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E8:64_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_1.html#E5:64_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:64_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span></span>)</span> <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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E10:64_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_1.html#E4:64_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E11:64_2"/>
( <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:64"/>
( <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> iff not <span class="p1"><a href="int_1.html#K1" target="_self">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1" target="_self">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2" target="_self">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2" target="_self">\]</a></span> )
 <b>by </b><i><a class="txt" href="int_1.html#E2:64"><i><font color="Green">E33</font></i></a></i>;<br/>


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