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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V2">anti-discrete</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:53"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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="E1:53_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:53_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:53"/>
<font color="Maroon">A</font> is    <a href="yellow13.html#M1" target="_self">basis</a> of <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2" target="_self">YELLOW13:def 2</a><br/>

<a NAME="E1:53_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<a NAME="E2:53_2"/><i><font color="Green">E36</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="tdlat_3.html#T20">TDLAT_3:20</a></i>;<br/>
<a NAME="E3:53_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_2_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:53_2_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:53_2"/>
( <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="yellow13.html#M1" target="_self">basis</a> of <font color="Maroon">p</font> ;<br/>
<a NAME="E6:53"/>
<font color="Maroon">A</font> is <a href="yellow13.html#V1" target="_self">correct</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D3" target="_self">YELLOW13:def 3</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_3_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:53_3_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:53_3_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</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></i>;<br/><a NAME="E4:53_3_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43">TOPS_1:43</a></i>;<br/><b>hence </b><a NAME="E5:53_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:53_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:53_3"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T20">TDLAT_3:20</a></i>;<br/>
<a NAME="E4:53_3"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_3_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:53_3_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:53_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span> is  <a href="yellow13.html#V1" target="_self">correct</a>  <a href="yellow13.html#M1" target="_self">basis</a> of <font color="Maroon">p</font>
 ;<br/>


</div>
