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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>
 ;<br/>

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E5:15"/><i><font color="Green">E46</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E6:15"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>;<br/>
<a NAME="E7:15"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> &amp;  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E8:15"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font> )
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E8:15_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E3:15_2"/><b>then </b>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:15_2"/>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:15_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:15_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:15_3_1"/><i><font color="Green">E67</font></i>: 
 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="relset_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</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="E4:15_3_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E5:15_3_1"/><b>then </b>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:15_3_1"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:15_3"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><a NAME="E5:15_3"/><b>then </b>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:15_3"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E12:15"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">T</font> is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i>, , , , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>

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

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font>, <font color="Maroon">IntA</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">cB</font> = <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E15:15"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E16:15"/><b>then </b>
<font color="Maroon">IntA</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E17:15"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">IntA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:15"/><b>then </b>
( <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">IntA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IntA</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E19:15"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E20:15"/>
<font color="Maroon">cB</font> <a href="hidden.html#R2">in</a>  <a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:15"/><b>then </b>
( <font color="Maroon">cB</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E22:15"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>


</div>
