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<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9
 ;<br/>

<b>set </b><font color="Maroon">n</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span>;<br/>
<a NAME="E2:71"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a>, </i>;<br/>
<a NAME="E4:71"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E5:71"/><b>then </b><i><font color="Green">E51</font></i>: 
2 <a href="power.html#K4">to_power</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/>
<a NAME="E6:71"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1"/>
( <font color="Maroon">x</font> is   <a href="int_1.html#NM1">Integer</a> or  not <font color="Maroon">x</font> is   <a href="int_1.html#NM1">Integer</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_1"/>
<font color="Maroon">x</font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/><div class="add"><a NAME="E2:71_1_1_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T59">INT_1:59</a></i>;<br/><b>hence </b><a NAME="E3:71_1_1_1"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2"/>
<font color="Maroon">x</font> is  not   <a href="int_1.html#NM1">Integer</a>
 ;<br/><div class="add"><a NAME="E2:71_1_1_2"/><b>then </b>
not <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T59">INT_1:59</a></i>;<br/><a NAME="E3:71_1_1_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</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">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T66">INT_1:66</a></i>;<br/><b>hence </b><a NAME="E4:71_1_1_2"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:71"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E8:71"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E9:71"/><b>then </b>
2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/>
<a NAME="E10:71"/><b>then </b><i><font color="Green">E52</font></i>: 
2 <a href="power.html#K4">to_power</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:71"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:71"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:71"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:71"/><b>then </b><i><font color="Green">E59</font></i>: 
2 <a href="power.html#K4">to_power</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:71"/><i><font color="Green">E60</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E16:71"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E17:71"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E2:71"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E18:71"/>
2 <a href="power.html#K4">to_power</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:71"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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