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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span> ;<br/>
<a NAME="E1:33"/><i><font color="Green">E38</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>by </b><i/>;<br/>
<a NAME="E4:33_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>by </b><i/>;<br/>
<a NAME="E5:33"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:33"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E7:33"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font> : ( <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:33"/>
<font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


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