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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:65"/><i><font color="Green">E42</font></i>: 
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">A</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">S</font> being 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#K1">REAL</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> holds <br/><font color="Olive">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Olive">S</font> );<br/>
<a NAME="E2:65"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E3:65"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E4:65"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:65"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:65_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:65_1"/><b>then </b>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:65_1"/><b>then </b>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font>
 
<b>by </b><i>, <a class="txt" href="topgen_5.html#E3:65"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:65"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:65"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E3:65_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>assume </b><a NAME="E4:65_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_2_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_1"><b>suppose </b></a><a NAME="E1:65_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:65_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_2_1_1_1"/>
not <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>take </b>
<font color="Maroon">fx</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be  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#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E2:65_2_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E3:65_2_1_1_1"/>
( <font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:65"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="funct_7.html#T6">FUNCT_7:6</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:65_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:65_2_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<b>hence </b><a NAME="E6:65_2_1_1_1"/>
<font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E7:65"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_5.html#E6:65"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_2"><b>suppose </b></a><a NAME="E1:65_2_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">f</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:65_2_1_2"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">S</font> being 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#K1">REAL</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Olive">S</font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T5" target="_self">Th5</a></i>;<br/><b>consider </b><font color="Maroon">h</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:65_2_1_2"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:65_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_2_1_2_1"/>
not <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be  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#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E2:65_2_1_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E3:65_2_1_2_1"/>
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E4:65_2_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font>, <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E4:65"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E5:65_2_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E6:65_2_1_2_1"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E6:65"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">cFB</font> =  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cFBa</font> = <font color="Maroon">cFB</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E9:65_2_1_2_1"/>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>;<br/>
<a NAME="E10:65_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cFB</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:65_2_1_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">cFBa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_5.html#T7" target="_self">Th7</a>, <a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>;<br/>
<a NAME="E12:65_2_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">cFBa</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S1</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">cFBa</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#K1">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E14:65_2_1_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">S1</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E7:65"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E15:65_2_1_2_1"/>
( <font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:65"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="funct_7.html#T6">FUNCT_7:6</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:65_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:65_2_1_2_1"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<b>then </b> <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">S</font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="pre_circ.html#NK1" target="_self">max</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="sprect_1.html#T53">SPRECT_1:53</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>
;<br/>
<b>hence </b><a NAME="E19:65_2_1_2_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:65"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, , );<br/></i>
<a NAME="E9:65"/>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:65"><i><font color="Green">E49</font></i></a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E10:65"/>
 ex <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">S</font> being 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#K1">REAL</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> holds <br/><font color="Olive">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Olive">S</font>
 <b>by </b><i/>;<br/>


</div>
