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<a NAME="E1:29_1"/><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">P<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="E3:29_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( 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="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:29_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:29_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1"/>
<font color="Maroon">P<sub>1</sub></font>[0]
 <b>by </b><i/>;<br/>


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