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<a NAME="E1:29_1"/><i><font color="Green">E16</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:29"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>(<a class="txt" href="nat_1.html#E1:29_1"><i><font color="Green">E16</font></i></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">c<sub>1</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_1.html#E2:29"><i><font color="Green">E15</font></i></a>, <a class="txt" href="nat_1.html#E3:29_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:29_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E3:29_1"><i><font color="Green">E17</font></i></a></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><a class="txt" href="nat_1.html#E3:29_1"><i><font color="Green">E17</font></i></a></i>;<br/>


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