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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>; <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   <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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b4">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Olive" title="b3">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/><b>let </b><font color="Maroon" title="c2">X0</font> be    <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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b3">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Olive" title="b2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b1">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E36">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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><br/>

<b>assume </b><a NAME="E2:56"/><i><font color="Green" title="E37">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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><br/>

<b>assume </b><a NAME="E3:56"/><i><font color="Green" title="E38">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_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) iff <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">P</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</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="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E5:56"/>
( <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_1.html#V1" title="TOPS_1:attr.1">dense</a> implies <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> or <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<a NAME="E2:56_1"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:56_1"/>
<font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/>

<a NAME="E4:56_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X0</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a></i>;<br/>
<a NAME="E5:56_1"/><b>then </b>
<font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="pre_topc.html#T47" title="PRE_TOPC:th.47">PRE_TOPC:47</a></i>;<br/>
<a NAME="E6:56_1"/><b>then </b><i><font color="Green" title="E40">A5</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:56_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E2:56_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E8:56_1"/><b>then </b>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="#E6:56_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:56_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E10:56_1"/>
<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="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</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="E6:56"/>
( <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )
 <b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E3:56"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="tops_3.html#T59" target="_self" title="TOPS_3:th.59">Th59</a>, <a class="ref" href="tops_1.html#T79" title="TOPS_1:th.79">TOPS_1:79</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>
