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<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>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">g</font> be   <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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E4:50"/>
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_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="E1:50_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:50_1_1"/>
<font color="Maroon">k</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:50_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:50_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:50_1_1"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <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 ) &amp; <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 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:50_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> &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> <font color="Maroon">g</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="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E9:50_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:50_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>


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

<a NAME="E5:50"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E6:50"/><b>then </b>
( ( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> 0 ) &amp; ( 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#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:50"/>
<font color="Maroon">g</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a>
 <b>by </b><i/>;<br/>

<a NAME="E8:50"/>
( <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<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="E10:50"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:50"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E12:50"/><b>then </b>
( <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 &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> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:50"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &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> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:50"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:50"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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="Maroon">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> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:50"/>
<font color="Maroon">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="Maroon">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#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/>


</div>
