<?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">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>,<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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1"/>
not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:52_1_1_1"/>
<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>  is <a href="xboole_0.html#V1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1_1_1_1"/>
not <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>  is <a href="xboole_0.html#V1">empty</a>
 ;<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="E3:52_1_1_1_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> &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> 
 <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">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="E5:52_1_1_1_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> &amp; <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> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:52_1_1_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:52_1_1_1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; 0 <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:52_1_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</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/>
<a NAME="E9:52_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</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="txt" href="stirl2_1.html#E1:52_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:52_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:52_1_1_1"/><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> <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="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, , </i>;<br/><b>hence </b><a NAME="E4:52_1_1_1"/>
 <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> &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="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> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:52_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> implies <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">i</font>,<font color="Olive">j</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></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="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> implies <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</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> ) );<br/><a NAME="E3:52_1_1_2"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E4:52_1_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>  st <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f'</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>;<br/>

<b>assume </b><a NAME="E1:52_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E2:52_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1_1_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>]
 ;<br/>

<a NAME="E2:52_1_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i/>;<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> <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="E4:52_1_1_2_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.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> &amp; ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 implies <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> ) )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">i</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E5:52_1_1_2_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E6:52_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:52_1_1_2_1_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:52_1_1_2_1_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">fn</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E11:52_1_1_2_1_1"/>
( <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font> &amp; <font color="Maroon">fi</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">fi</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> &amp; ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> implies <font color="Maroon">fi</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</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><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, <a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/>
<b>hence </b><a NAME="E12:52_1_1_2_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</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="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> implies <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</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> ) )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:52_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">fn</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font><b> such that </b><br/><a NAME="E3:52_1_1_2_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:52_1_1_2_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">fn</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">fn</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>and </b><br/><a NAME="E5:52_1_1_2_1_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">fn</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</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> )
 ;<br/>
<b>let </b><font color="Maroon">i</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E6:52_1_1_2_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E7:52_1_1_2_1_2"/>
( <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f'</font> as   <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 ;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:52_1_1_2_1_2"/>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E11:52_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f1</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> &amp; ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> implies <font color="Maroon">f1</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</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>, , <a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E12:52_1_1_2_1_2"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</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="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> implies <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</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> ) )
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>set </b><font color="Maroon">FF1</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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">FF2</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">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </span>}</span> ;<br/><a NAME="E5:52_1_1_2"/><i><font color="Green">E77</font></i>: 
 <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="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </span>}</span>  <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> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </span>}</span> 
 <b>from </b><i><a class="ref" href="stirl2_1.html#S4" target="_self">STIRL2_1:sch 4</a>(<a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, );<br/></i><a NAME="E6:52_1_1_2"/><i><font color="Green">E78</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="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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1_1_2_2"/>
<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> <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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_2_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="E1:52_1_1_2_2_1"/><i><font color="Green">E79</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/>

<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="E3:52_1_1_2_2_1"/><i><font color="Green">E80</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> &amp; <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> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:52_1_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_2_1_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:52_1_1_2_2_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E2:52_1_1_2_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> ;<br/>
<a NAME="E4:52_1_1_2_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E5:52_1_1_2_2_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">i</font>,<font color="Maroon">j</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> implies <font color="Olive">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</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><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:52_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, , , </i>;<br/>
<b>hence </b><a NAME="E6:52_1_1_2_2_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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:52_1_1_2_2"/>
<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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_2_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="E1:52_1_1_2_2_2"/><i><font color="Green">E104</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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) </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="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E3:52_1_1_2_2_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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>,<font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>] &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:52_1_1_2_2_2"/>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</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; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:52_1_1_2_2_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="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Olive">f'</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="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 implies <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><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1_1_2_2_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> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E7:52_1_1_2"/>
<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>  <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>,<font color="Maroon">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1_1_2_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>,<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>  <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>,<font color="Maroon">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_3_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="E1:52_1_1_2_3_1"/><i><font color="Green">E107</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">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> 
 ;<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">k</font><b> such that </b><br/><a NAME="E3:52_1_1_2_3_1"/><i><font color="Green">E108</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><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:52_1_1_2_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>]
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1_1_2_3_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>,<font color="Maroon">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:52_1_1_2_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>,<font color="Maroon">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </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>,<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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_3_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="E1:52_1_1_2_3_2"/><i><font color="Green">E109</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">k</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] </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">k</font><b> such that </b><br/><a NAME="E3:52_1_1_2_3_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>] )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:52_1_1_2_3_2"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Olive">g</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">g</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> &amp; ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Olive">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</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><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1_1_2_3_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>,<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><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E8:52_1_1_2"/>
 <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> &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="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>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:52"/>
 <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> &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="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> 
 ;<br/>


</div>
