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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font> be    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E1:69"/><i><font color="Green" title="E54">A1</font></i>: 
 ex <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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" title="c1">n</font>
;<br/>

<b>thus </b><a NAME="E1:69_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 <b>by </b><i><a class="ref" href="comptrig.html#D2" target="_self" title="COMPTRIG:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E2:69"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<div><i><font color="Green" title="E56">A3</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" title="c3">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:69_2"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:69_2"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:69_2"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">t</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="comptrig.html#E1:69_2"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="comptrig.html#E4:69_2"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">t</font>;<br/><b>thus </b><a NAME="E6:69_2"/>
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E4:69_2"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:69_2"/>
<font color="Maroon" title="c2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E4:69_2"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><b>hence </b><a NAME="E8:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">t</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="comptrig.html#E2:69_2"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">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" title="COMPTRIG:sch.1">COMPTRIG:sch 1</a>(<a class="txt" href="comptrig.html#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="comptrig.html#E3:69"><i><font color="Green" title="E56">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:69"/>
contradiction
 <b>by </b><i><a class="txt" href="comptrig.html#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>


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