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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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#V3" title="STRUCT_0:attr.3">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="E1:64"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:64"/>
<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>then reconsider </b><font color="Maroon" title="c6">E</font> = <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> ;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c6">E</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:64"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="tops_2.html#T33" title="TOPS_2:th.33">TOPS_2:33</a></i>;<br/>
<b>assume </b><a NAME="E5:64"/><i><font color="Green" title="E40">A3</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="E6:64"/><i><font color="Green" title="E41">A4</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/>

<a NAME="E7:64"/><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="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</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="c7">D</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</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="txt" href="tops_3.html#E6:64"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:64"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</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="c6">E</font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E5:64"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="tops_3.html#E6:64"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="tops_3.html#E4:64"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E11:64"/><b>then </b>
<font color="Maroon" title="c7">D</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:64"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="tsep_1.html#T9" title="TSEP_1:th.9">TSEP_1:9</a></i>;<br/>
<a NAME="E12:64"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c7">D</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="c4">A</font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E6:64"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="tops_3.html#E7:64"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></i>;<br/>
<b>thus </b><a NAME="E13:64"/>
( <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:64_1"/>
<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="E3:64_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></i>;<br/>
<a NAME="E4:64_1"/><b>then </b>
<font color="Maroon" title="c7">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E10:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="tops_3.html#E1:64_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="tops_3.html#T60" target="_self" title="TOPS_3:th.60">Th60</a></i>;<br/>
<a NAME="E5:64_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E12:64"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="tops_1.html#T79" title="TOPS_1:th.79">TOPS_1:79</a></i>;<br/>
<b>hence </b><a NAME="E6:64_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="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></i>;<br/>


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

<b>thus </b><a NAME="E14:64"/>
( <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_1.html#V1" title="TOPS_1:attr.1">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> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
<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="txt" href="tops_3.html#E5:64"><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></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T33" target="_self" title="TOPS_3:th.33">Th33</a></i>;<br/>

<b>thus </b><a NAME="E4:64_2"/>
<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#E6:64"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="tops_3.html#E1:64_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="tops_3.html#T61" target="_self" title="TOPS_3:th.61">Th61</a></i>;<br/>


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


</div>
