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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:23_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:23_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>
<a NAME="E4:23_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:23_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:23_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:23_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E10:23_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:23_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 0</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E12:23_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


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

<b>thus </b><a NAME="E6:23_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


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