<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">n</font> &amp; 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/>
<span class="lab"><font color="Green" title="E5">A2</font></span>: 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>

<br/>.= 
<font color="Maroon" title="c1">s</font>

;<br/>
<span class="kw">assume </span><a NAME="E4:3"/>
<font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">n</font> &amp; 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> )</span><br/>

<a NAME="E5:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A3</font></span>: 
1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">n</font> &amp; 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> )</span><br/>

<a NAME="E6:3"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3"/>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">n</font> &amp; 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="xreal_1.html#T87" title="XREAL_1:th.87">XREAL_1:87</a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
