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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="comptrig.html#M1" target="_self">CRoot</a> of <font color="Maroon">n</font>,0;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">v</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E1:69"/><i><font color="Green">E28</font></i>: 
 ex <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E1:69_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E25"><i><font color="Green">Lemma54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E2:69"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:69_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:69_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:69_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="comptrig.html#E29"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">t</font> = <font color="Maroon">t</font>;<br/><b>thus </b><a NAME="E6:69_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&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/><a NAME="E7:69_2"/>
<font color="Maroon">v</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="newton.html#K2">|^</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>hence </b><a NAME="E8:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E27"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="comptrig.html#T22" target="_self">Th22</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:69"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>from </b><i><a class="ref" href="comptrig.html#S1" target="_self">COMPTRIG:sch 1</a>(<a class="txt" href="comptrig.html#E26"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="comptrig.html#E28"><i><font color="Green">Lemma57</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:69"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a>, <a class="txt" href="comptrig.html#E27"><i><font color="Green">Lemma56</font></i></a></i>;<br/>


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