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

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <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_3.html#V1" title="TOPS_3:attr.1">everywhere_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/><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <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_3.html#V1" title="TOPS_3:attr.1">everywhere_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/><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <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_3.html#V1" title="TOPS_3:attr.1">everywhere_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/><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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_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> ) 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/>







<b>assume </b><a NAME="E1:60"/><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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_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> ) 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/>

<b>assume </b><a NAME="E2:60"/><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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_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> ) 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/>

<b>assume </b><a NAME="E3:60"/><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <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> ) 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/>

<b>thus </b><a NAME="E4:60"/>
( <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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_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> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<a NAME="E2:60_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_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="E3:60_1"/><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="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="txt" href="#E1:60_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="tops_3.html#T32" target="_self" title="TOPS_3:th.32">Th32</a>, <a class="ref" href="tops_3.html#T33" target="_self" title="TOPS_3:th.33">Th33</a></i>;<br/>
<a NAME="E4:60_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="#E1:60"><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="E6:60_1"/><i><font color="Green" title="E44">A7</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E7:60_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="#E2:60_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="tops_3.html#T61" target="_self" title="TOPS_3:th.61">Th61</a></i>;<br/>
<a NAME="E8:60_1"/><b>then </b><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> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="#E6:60_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="tops_3.html#T44" target="_self" title="TOPS_3:th.44">Th44</a></i>;<br/>
<a NAME="E9:60_1"/><i><font color="Green" title="E46">A9</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="E11:60_1"/>
<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="#E3:60_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="#E8:60_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="#E9:60_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="tops_3.html#T62" target="_self" title="TOPS_3:th.62">Th62</a></i>;<br/>
<b>hence </b><a NAME="E12:60_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="#E3:60"><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="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E5:60"/>
( <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="#E2:60"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E3:60"><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
