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<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>for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1;<br/>
<a NAME="E1:25_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:25_1"/><i><font color="Green" title="E10">A3</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <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">k</font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:25"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 <b>and </b><br/><a NAME="E5:25_1"/><i><font color="Green" title="E12">A9</font></i>: 
for <font color="Olive" title="b1">m</font> being  <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">m</font>] holds <br/><font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="nat_1.html#E2:25_1"><i><font color="Green" title="E10">A3</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">k</font>
;<br/>

<b>thus </b><a NAME="E6:25_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 <b>by </b><i><a class="txt" href="nat_1.html#E2:25"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<a NAME="E4:25_1_1"/>
( <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">k</font> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="nat_1.html#E4:25_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="nat_1.html#E1:25_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="nat_1.html#E3:25_1_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<a NAME="E5:25_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self" title="NAT_1:th.2">Th2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:25_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</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" target="_self" title="NAT_1:th.6">Th6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/><a NAME="E2:25_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="nat_1.html#E4:25_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="nat_1.html#E1:25_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E7:25_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="nat_1.html#T8" target="_self" title="NAT_1:th.8">Th8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:25_1_1"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="nat_1.html#E5:25_1"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<a NAME="E10:25_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E11:25_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E7:25_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="nat_1.html#E9:25_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:25_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><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">k</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E4:25_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>


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