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





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:26"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">k</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/><b>consider </b><font color="Maroon">l'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">l'</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">l'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">m'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:26_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">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="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:26_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="funct_1.html#T52">FUNCT_1:52</a></i>;<br/><a NAME="E7:26_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <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"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <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"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <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"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <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"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_1.html#T144">FUNCT_1:144</a></i>;<br/><a NAME="E8:26_1"/><b>then </b>
( <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"><font color="Maroon">l</font></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">l</font></span><a href="domain_1.html#K6">}</a></span> &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"><font color="Maroon">m</font></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">m</font></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="E9:26_1"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</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="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</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="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:26_1"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</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="Maroon">l</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">f</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="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
<font color="Maroon">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a>
 <b>by </b><i>, </i>;<br/>


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