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<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>, <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>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</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">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #)
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><br/> for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">Y</font> is <a href="compts_1.html#V6">compact</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9">WAYBEL_3:def 9</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon">X</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="E3:42"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:42"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="tops_3.html#T76">TOPS_3:76</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:42"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> is <a href="compts_1.html#V6">compact</a> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <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/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E10:42"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> is <a href="compts_1.html#V6">compact</a> )
 <b>by </b><i>, , , <a class="ref" href="tops_3.html#T77">TOPS_3:77</a></i>;<br/>


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