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

<b>let </b><font color="Maroon">X</font> be   <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">f</font> be  <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>, <font color="Maroon">g</font> be  <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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:64"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1">SUBSET_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1">SUBSET_1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
( <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> or ( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> ) or ( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> ) or ( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><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>;<br/><b>take </b>
<font color="Maroon">h</font>
;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><a NAME="E3:64_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:64_1_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:64_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">g</font>
;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><a NAME="E2:64_1_2"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:64_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:64_1_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_3"><b>suppose </b></a><a NAME="E1:64_1_3"/><i><font color="Green">E55</font></i>: 
( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">f</font>
;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><a NAME="E2:64_1_3"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:64_1_3"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:64_1_3"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4"><b>suppose </b></a><a NAME="E1:64_1_4"/><i><font color="Green">E56</font></i>: 
( not <font color="Maroon">X</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/><a NAME="E3:64_1_4"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X'</font> holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> ) &amp; ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> ) &amp; ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> <b>by </b><i><a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="txt" href="topgen_5.html#E4:64"><i><font color="Green">E43</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font>, <font color="Maroon">gg</font> = <font color="Maroon">g</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> ;<br/><a NAME="E6:64_1_4"/><i><font color="Green">E61</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">gg</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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:64_1_4"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A'</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A'</font></span>)</span>

<br/>.= 
<font color="Maroon">A'</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A'</font></span>)</span>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T33">RELSET_1:33</a></i>;<br/>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B'</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B'</font></span>)</span>

<br/>.= 
<font color="Maroon">B'</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B'</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B'</font></span>)</span>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T33">RELSET_1:33</a></i>;<br/><a NAME="E12:64_1_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:64_1_4_3"/>
<font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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/>

<a NAME="E2:64_1_4_3"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> )
 
;<br/>
<a NAME="E3:64_1_4_3"/><b>then </b>
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:64_1_4_3"/>
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E79</font></i>: <font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span> = 
<font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span>

<br/>.= 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #)
<b>by </b><i><a class="ref" href="tsep_1.html#T100">TSEP_1:100</a></i>
;<br/><a NAME="E14:64_1_4"/><i><font color="Green">E80</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E15:64_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">f'</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4_5"><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="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:64_1_4_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g'</font></span>)</span>
 ;<br/>

<a NAME="E2:64_1_4_5"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:64_1_4_5"/><b>then </b>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:64_1_4_5"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><font color="Maroon">f'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="topgen_5.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="topgen_5.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E16:64_1_4"/><b>then </b>
( <font color="Maroon">f'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g'</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span></span>)</span>,<font color="Maroon">R</font> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font> #) )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g'</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>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/><b>take </b>
<font color="Maroon">h</font>
;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><a NAME="E18:64_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> ) )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:64_1_4"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T16">FUNCT_4:16</a></i>;<br/><b>hence </b><a NAME="E20:64_1_4"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
