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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:8"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">n</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E5:8"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E6:8"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E8:8"/><b>then </b><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></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/>
<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="int_1.html#K1">/]</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="E10:8"/><i><font color="Green">E41</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:8_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E3:8_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:8_1"/>
<font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:8_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">N</font> = <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E7:8_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">N</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><b>hence </b><a NAME="E8:8_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="topreala.html#T2">TOPREALA:2</a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<a NAME="E12:8"/>
( <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E13:8"/><b>then </b>
( <font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a>, <a class="ref" href="xreal_1.html#T83">XREAL_1:83</a></i>;<br/>
<b>hence </b><a NAME="E14:8"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T81">XREAL_1:81</a></i>;<br/>


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