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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">k</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:26"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">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" title="c4">l</font>, <font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c6">l'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c6">l'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:26_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">m'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:26_1"/><i><font color="Green" title="E21">A6</font></i>: 
( <font color="Maroon" title="c7">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c7">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:26_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:26_1"/>
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> &amp; <font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m'</font> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E3:26_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E5:26_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:26_1"/><b>then </b>
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:26_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="stirl2_1.html#E3:26_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E5:26_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T144" title="FUNCT_1:th.144">FUNCT_1:144</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E8:26_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:26_1"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self" title="STIRL2_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E10:26_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:26_1"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
<font color="Maroon" title="c3">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E4:26"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="stirl2_1.html#D1" target="_self" title="STIRL2_1:def.1">Def1</a></i>;<br/>


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