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

<b>let </b><font color="Maroon">T1</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">S</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">T2</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">T</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">F2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">T1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">T2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> )
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E6:7"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:7"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:7"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 ;<br/>

<b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<i><font color="Green">E49</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<b>then </b><i><font color="Green">E50</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E12:7"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E13:7"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></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="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E15:7"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E16:7"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:7_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:7_4"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<b>then </b><i><font color="Green">E54</font></i>: <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E4:7_4"/><i><font color="Green">E55</font></i>: 
for <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>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_4_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_4_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_4_2_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font>
 ;<br/><div class="add"><a NAME="E2:7_4_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:7_4_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:7_4_2_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_4_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_4_2_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_4_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:7_4_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>
 ;<br/>

<a NAME="E2:7_4_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:7_4_3_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</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="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:7_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:7_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E3:7_4_3"/><b>then </b>
( <font color="Maroon">x</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="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:7_4_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:7_4_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:7_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:7_4"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:7_4_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_4_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span>
 ;<br/>

<a NAME="E2:7_4_4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:7_4_4_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E4:7_4_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:7_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E3:7_4_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:7_4_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:7_4_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:7_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:7_4"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E9:7_4"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 
;<br/>
<a NAME="E10:7_4"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:7_4"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E13:7_4"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
 
;<br/>
<a NAME="E14:7_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E15:7_4"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon">P3</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">P4</font> = <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>;<br/>
<a NAME="E17:7_4"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, , , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">F01</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E19:7_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">F01</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F01</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E20:7_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, , <a class="ref" href="jgraph_2.html#T5" target="_self">Th5</a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<b>consider </b><font color="Maroon">F02</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E22:7_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">F02</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F02</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E23:7_4"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, , <a class="txt" href="jgraph_2.html#E1:7"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E24:7_4"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:7"/>
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div>
