<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X0</font> being   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/>  for <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X0</font>  st <font color="Olive" title="b2">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">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="Olive" title="b1">X0</font> &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B</font> holds <br/>( ( <font color="Olive" title="b2">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b4">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) iff <font color="Olive" title="b3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> holds <br/>( ( <font color="Olive" title="b1">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b3">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) iff <font color="Olive" title="b2">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Maroon" title="c3">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <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> &amp; <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> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/>( ( <font color="Maroon" title="c3">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <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> &amp; <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> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> 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>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <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> or  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> or ( ( <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>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:58"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> or ( ( <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>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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="E4:58"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c6">E</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="tops_2.html#T25" title="TOPS_2:th.25">TOPS_2:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:58"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> or ( ( <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>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E6:58"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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>  ) iff <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>

<a NAME="E7:58"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E9:58"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="tops_1.html#T19" title="TOPS_1:th.19">TOPS_1:19</a></span>;<br/>
<a NAME="E10:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="tops_1.html#T23" title="TOPS_1:th.23">TOPS_1:23</a></span>;<br/>
<a NAME="E11:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c7">D</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="tsep_1.html#T9" title="TSEP_1:th.9">TSEP_1:9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:58"/>
( <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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  or <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:58_1"/>
<font color="Maroon" title="c5">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/>

<a NAME="E3:58_1"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></span>;<br/>
<a NAME="E4:58_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:58"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="tops_3.html#T60" target="_self" title="TOPS_3:th.60">Th60</a></span>;<br/>
<a NAME="E5:58_1"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="tops_1.html#T24" title="TOPS_1:th.24">TOPS_1:24</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:58_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E13:58"/>
( <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>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:58_2"/>
<font color="Maroon" title="c3">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="tops_3.html#T33" target="_self" title="TOPS_3:th.33">Th33</a>, <a class="ref" href="tops_3.html#T38" target="_self" title="TOPS_3:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:58_2"/>
<font color="Maroon" title="c5">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="tops_3.html#T61" target="_self" title="TOPS_3:th.61">Th61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
