<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">k</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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>set </b><font color="Maroon">F1</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">F2</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E1:74"/><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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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">{<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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K2">\/</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="E1:74_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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1<b> such that </b><br/><a NAME="E3:74_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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="E4:74_1"/>
( <font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<a NAME="E5:74_1"/><b>then </b>
( <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  or <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E6:74_1"/>
<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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K2">\/</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:74"/>
<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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K2">\/</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><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="E1:74_2"/><i><font color="Green">E47</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K2">\/</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:74_2"/>
( <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  or <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:74_2"/><b>then </b>
(  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) or  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) )
 
;<br/>
<b>hence </b><a NAME="E4:74_2"/>
<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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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/>


</div><b>end;</b></div>
<a NAME="E3:74"/><b>then </b><i><font color="Green">E58</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K2">\/</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="hidden.html#R1">=</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green">E62</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#R1">misses</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_3"/>
<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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#NR5" target="_self">meets</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:74_3"/><i><font color="Green">E63</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E4:74_3"/>
( <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  &amp; <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:74_3"/><b>then </b>
(  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) &amp;  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) )
 
;<br/>
<b>hence </b><a NAME="E6:74_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:74"/><i><font color="Green">E65</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_4"><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="E1:74_4"/><i><font color="Green">E67</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:74_4"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:74_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:74"/><i><font color="Green">E68</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5"><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="E1:74_5"/><i><font color="Green">E69</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:74_5"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:74_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:74"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="fraenkel.html#T16">FRAENKEL:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F1</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> , <font color="Maroon">F2</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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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> &amp; <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, , <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E9:74"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <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">k</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="E10:74"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <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> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 : ( <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">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E11:74"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T14" target="_self">Th14</a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>


</div>
