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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>;<br/>







<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ;<br/>

<b>assume </b><a NAME="E2:66"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 ;<br/>

<b>assume </b><a NAME="E3:66"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 ;<br/>

<b>thus </b><a NAME="E4:66"/>
( <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> implies <font color="Maroon" title="c4">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T32" target="_self" title="TOPS_3:th.32">Th32</a></i>;<br/>
<a NAME="E3:66_1"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_3.html#T33" target="_self" title="TOPS_3:th.33">Th33</a></i>;<br/>
<a NAME="E4:66_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">D</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font> <b>by </b><i><a class="txt" href="tops_3.html#E1:66"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E6:66_1"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c5">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<a NAME="E7:66_1"/>
<font color="Maroon" title="c6">D</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:66_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="tops_3.html#T61" target="_self" title="TOPS_3:th.61">Th61</a></i>;<br/>
<a NAME="E8:66_1"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E6:66_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="tops_3.html#T44" target="_self" title="TOPS_3:th.44">Th44</a></i>;<br/>
<a NAME="E9:66_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">E</font> = <font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:66_1"/><i><font color="Green" title="E46">A9</font></i>: 
<font color="Maroon" title="c7">E</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:66_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="tops_3.html#E8:66_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="tops_3.html#E9:66_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="tops_3.html#T62" target="_self" title="TOPS_3:th.62">Th62</a></i>;<br/>
<a NAME="E12:66_1"/>
<font color="Maroon" title="c7">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:66"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E13:66_1"/>
<font color="Maroon" title="c4">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E11:66_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="tops_3.html#T38" target="_self" title="TOPS_3:th.38">Th38</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:66"/>
( <font color="Maroon" title="c4">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> implies ( <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> ) )
 <b>by </b><i><a class="txt" href="tops_3.html#E2:66"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="tops_3.html#E3:66"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="tops_3.html#T38" target="_self" title="TOPS_3:th.38">Th38</a>, <a class="ref" href="tops_3.html#T61" target="_self" title="TOPS_3:th.61">Th61</a></i>;<br/>


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