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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">PT</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">PT</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  implies  for <font color="Olive" title="b1">FX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">FX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Olive" title="b1">FX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/> ex <font color="Olive" title="b2">HX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b2">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b3">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">HX</font> holds <br/> ex <font color="Olive" title="b4">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b4">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">W</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c1">PT</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">FX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">FX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Olive" title="b1">FX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/> ex <font color="Olive" title="b2">HX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b2">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b3">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">HX</font> holds <br/> ex <font color="Olive" title="b4">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b4">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">W</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">FX</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">FX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Maroon" title="c2">FX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  implies  ex <font color="Olive" title="b1">HX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">HX</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b3">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">W</font> ) ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c2">FX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font>
 <span class="kw">and </span><br/><a NAME="E3:6"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c2">FX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">HX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">HX</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b3">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">W</font> ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">V1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) );<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">HX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font><span class="kw"> such that </span><br/><a NAME="E5:6"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> holds <br/> ( <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">V</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" onmouseover="rs('subset_1/S3')" onmouseout="rh()">SUBSET_1:sch 3</a>();<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">HX</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c3">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) ) )</span><br/>

<a NAME="E6:6"/>
 for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> holds <br/><font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> implies <font color="Maroon" title="c4">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1"/>
<font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Olive" title="b1">V1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:6_1"/>
<font color="Maroon" title="c4">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E7:6"/>
<font color="Maroon" title="c3">HX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; (  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) ) )</span><br/>

<a NAME="E8:6"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">PT</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">PT</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">PT</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font><span class="kw"> such that </span><br/><a NAME="E4:6_2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">and </span><br/><a NAME="E5:6_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="pcomps_1.html#T3" onmouseover="rs('pcomps_1/T3')" onmouseout="rh()">PCOMPS_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">V</font> = <font color="Maroon" title="c6">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( <font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:6_2_1_1_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font></span><br/><div class="add"><a NAME="E2:6_2_1_1_1"/>
<font color="Maroon" title="c7">V</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="#E3:6"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:6_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>;<br/><a NAME="E3:6_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ;<br/><a NAME="E4:6_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">X</font>, <font color="Maroon" title="c9">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font><span class="kw"> such that </span><br/><a NAME="E6:6_2_1_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c8">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E7:6_2_1_1_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c9">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E8:6_2_1_1_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">and </span><br/><a NAME="E9:6_2_1_1_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Y</font>
 <span class="kw">and </span><br/><a NAME="E10:6_2_1_1_1"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E1:6_2_1_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E3:6_2_1_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="compts_1.html#D2" onmouseover="rs('compts_1/D2')" onmouseout="rh()">COMPTS_1:def 2</a></span>;<br/><a NAME="E11:6_2_1_1_1"/>
( <font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6_2_1_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E10:6_2_1_1_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="subset_1.html#T23" onmouseover="rs('subset_1/T23')" onmouseout="rh()">SUBSET_1:23</a></span>;<br/><a NAME="E12:6_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A14</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T5" onmouseover="rs('tops_1/T5')" onmouseout="rh()">TOPS_1:5</a></span>;<br/><a NAME="E13:6_2_1_1_1"/>
<font color="Maroon" title="c9">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6_2_1_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="subset_1.html#T17" onmouseover="rs('subset_1/T17')" onmouseout="rh()">SUBSET_1:17</a></span>;<br/><a NAME="E14:6_2_1_1_1"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6_2_1_1_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a></span>;<br/><a NAME="E15:6_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:6_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:6_2_1_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:6_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2_1_1_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:6_2_1_1_2"/><span class="lab"><font color="Green" title="E9">A15</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font><span class="kw"> such that </span><br/><a NAME="E3:6_2_1_1_2"/><span class="lab"><font color="Green" title="E10">A16</font></span>: 
<font color="Maroon" title="c8">X</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">PT</font>
 ;<br/><a NAME="E4:6_2_1_1_2"/><span class="lab"><font color="Green" title="E11">A17</font></span>: 
<font color="Maroon" title="c8">X</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="#E3:6_2_1_1_2"><span class="lab"><font color="Green" title="E10">A16</font></span></a></span>;<br/><font color="Maroon" title="c7">V</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">PT</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_2_1_1_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">PT</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T6" onmouseover="rs('pre_topc/T6')" onmouseout="rh()">PRE_TOPC:6</a></span>
;<br/><a NAME="E6:6_2_1_1_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2_1_1_2"><span class="lab"><font color="Green" title="E10">A16</font></span></a>, <a class="ref" href="tops_1.html#T2" onmouseover="rs('tops_1/T2')" onmouseout="rh()">TOPS_1:2</a></span>;<br/><a NAME="E7:6_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:6_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:6_2_1_1_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:6_2_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2_1_1_2"><span class="lab"><font color="Green" title="E10">A16</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:6_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">HX</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E9:6"/>
<font color="Maroon" title="c3">HX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D11" onmouseover="rs('setfam_1/D11')" onmouseout="rh()">SETFAM_1:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E10:6"/>
<font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> )</span><br/>

<a NAME="E11:6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Olive" title="b1">V1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:6"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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