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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of 0,<font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span> ;<br/>
<a NAME="E2:34"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of 0,<font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of 0,<font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of 0,<font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of 0,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:34_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:34_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="stirl2_1.html#E5:34_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
0 <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>


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