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<div class="add">

<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E47</font></i>: 
( ( for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> ) &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>take </b><font color="Maroon">k</font> = 0;<br/><a NAME="E2:4_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E3:4_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="henmodel.html#K1" target="_self">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> </span>}</span> ;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> 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#T26">FINSET_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:4_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1" target="_self">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:4_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><a NAME="E6:4_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:4_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E10:4_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">a</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E12:4_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:4_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">d</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="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:4_2_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_2_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/><b>thus </b><a NAME="E4:4_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:4_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1" target="_self">min*</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E13:4_2_2"/>
( <font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">Y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:4_2_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/>
<a NAME="E16:4_2_2"/>
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E17:4_2_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> or <font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E18:4_2_2"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2_2_2"/>
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:4_2_2_2"/><b>then </b>
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><a NAME="E3:4_2_2_2"/><b>then </b>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E4:4_2_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Y'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:4_2_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:4_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i/>;<br/>


</div>
