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<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">L</font> be   <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font>;<br/>









<b>assume </b><a NAME="E1:24"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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">L</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font> #)
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><br/>

<b>assume </b><a NAME="E4:24"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E5:24"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/> ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</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="E1:24_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:24_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E4:24_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E5:24_1"/><b>then </b>
<font color="Maroon">S'</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:24_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S'</font> )
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E9:24_1"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:24"/>
<font color="Maroon">X</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">t</font>
 <b>by </b><i>, , , , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>


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