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<a NAME="E1:14"/>
( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c1">K</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3<b> such that </b><br/><a NAME="E3:14"/><i><font color="Green" title="E21">A1</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/>
<a NAME="E4:14"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">L</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 3
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">L</font> </span>}</span>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 3 )</span><br/>

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">L</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 3</span><br/>

<b>then consider </b><font color="Maroon" title="c3">L</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3<b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> &amp; 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">L</font> )
 ;<br/>
<b>thus </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 3
 <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E22">A2</font></i></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>then reconsider </b><font color="Maroon" title="c2">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">L</font> </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">K</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of 3 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #)
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<a NAME="E6:14"/>
3 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 3
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:14"/>
(  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp;  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 3,<font color="Maroon" title="c2">B</font> #) is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )
 <b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pencil_1.html#T8" target="_self" title="PENCIL_1:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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