<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:55_1"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>]
 <b>by </b><i><a class="txt" href="frechet2.html#E1:55"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<a NAME="E3:55_1"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T69" title="CARD_1:th.69">CARD_1:69</a></i>;<br/>
<a NAME="E4:55_1"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T52" title="CARD_1:th.52">CARD_1:52</a></i>;<br/>
<a NAME="E5:55_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E6:55_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">X</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="frechet2.html#E4:55_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/>
<a NAME="E8:55_1"/><i><font color="Green" title="E46">A4</font></i>: 
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">mm</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c2">X</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E10:55_1"/>
<font color="Maroon" title="c3">mm</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">mX</font> = <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">mX</font>
;<br/>

<b>let </b><font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E12:55_1"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c4">mX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ;<br/>

<a NAME="E13:55_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E14:55_1"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c5">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">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/>
<a NAME="E15:55_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:55_1_1"/><b>then </b>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> </span>}</span> 
 
;<br/>
<a NAME="E3:55_1_1"/><b>then </b>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>
<a NAME="E4:55_1_1"/><b>then </b>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/>
<a NAME="E5:55_1_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E14:55_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:55_1_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/>
<a NAME="E7:55_1_1"/><b>then </b><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E8:55_1_1"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c2">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet2.html#E8:55_1"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:55_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E12:55_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="frechet2.html#E7:55_1_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:55_1"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:55_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<a NAME="E17:55_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E18:55_1"/><b>then </b>
<font color="Maroon" title="c5">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">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:55_1"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="frechet2.html#K1" title="FRECHET2:func.1">*</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">m</font>]
 <b>by </b><i><a class="txt" href="frechet2.html#E16:55_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>


</div>
