<?xml version="1.0"?>
<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">  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="Maroon" title="c1">PT</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  &amp; <font color="Maroon" title="c1">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>  &amp; <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">GX</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">GX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b2">GX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp;  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Olive" title="b2">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">FX</font> &amp; <font color="Olive" title="b2">GX</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>  )</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="c1">PT</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  &amp; <font color="Maroon" title="c1">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>  &amp; <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">GX</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">GX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">GX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp;  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Olive" title="b1">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Olive" title="b1">GX</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c1">PT</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c1">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a> 
 <span class="kw">and </span><br/><a NAME="E3:7"/><span class="lab"><font color="Green" title="E5">A3</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> &amp; <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">GX</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">GX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">GX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp;  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Olive" title="b1">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Olive" title="b1">GX</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>  )</span><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="E5:7"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E6:7"/><span class="lab"><font color="Green" title="E7">A5</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>  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="pcomps_2.html#T2" target="_self" onmouseover="rs('pcomps_2/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="pcomps_1.html#T24" onmouseover="rs('pcomps_1/T24')" onmouseout="rh()">PCOMPS_1:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">GX</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="E8:7"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
( <font color="Maroon" title="c4">GX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c4">GX</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="E9:7"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c4">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">and </span><br/><a NAME="E10:7"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c4">GX</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="pcomps_1.html#D3" onmouseover="rs('pcomps_1/D3')" onmouseout="rh()">PCOMPS_1:def 3</a></span>;<br/>
<a NAME="E11:7"/><span class="lab"><font color="Green" title="E11">A9</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>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">GX</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> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">GX</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="c5">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="E1:7_1"/>
<font color="Maroon" title="c5">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">GX</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="c5">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">then </span><span class="kw">consider </span><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:7_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">HX</font>
 <span class="kw">and </span><br/><a NAME="E4:7_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7"><span class="lab"><font color="Green" title="E9">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">W</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="E7:7_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
( <font color="Maroon" title="c8">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="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">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"> ( <font color="Maroon" title="c8">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="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )</span><br/>

<a NAME="E8:7_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">V</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="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="pre_topc.html#T19" onmouseover="rs('pre_topc/T19')" onmouseout="rh()">PRE_TOPC:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:7_1"/>
( <font color="Maroon" title="c8">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="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1"><span class="lab"><font color="Green" title="E14">A12</font></span></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>
<a NAME="E12:7"/>
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c4">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="setfam_1.html#D2" onmouseover="rs('setfam_1/D2')" onmouseout="rh()">SETFAM_1:def 2</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="c5">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c4">GX</font> or  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c4">GX</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">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PT</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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="E4:7_2"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">and </span><br/><a NAME="E5:7_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">GX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">W</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="E7:7_2"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
( <font color="Maroon" title="c8">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="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E5:7_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">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"> ( <font color="Maroon" title="c8">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:7_2"/>
( <font color="Maroon" title="c8">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E7:7_2"><span class="lab"><font color="Green" title="E15">A16</font></span></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>
<span class="kw">hence </span><a NAME="E13:7"/>
 ex <font color="Olive" title="b1">GX</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">GX</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">GX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp;  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Olive" title="b1">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Olive" title="b1">GX</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E10:7"><span class="lab"><font color="Green" title="E10">A8</font></span></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>
