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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 3 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> implies for <font color="Olive" title="b1">S</font> being   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">L</font> where <font color="Olive" title="b2">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">L</font> </span>}</span>  holds <br/>( not <font color="Olive" title="b1">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> ) )</span><br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E18">A1</font></i>: 
3 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <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">S</font> being   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">L</font> where <font color="Olive" title="b2">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">L</font> </span>}</span>  holds <br/>( not <font color="Olive" title="b1">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Olive" title="b1">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<a NAME="E2:11"/><i><font color="Green" title="E19">B2</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">S</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <font color="Maroon" title="c1">X</font> : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L</font> </span>}</span>  implies ( not <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E21">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green" title="E22">A4</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <font color="Maroon" title="c1">X</font> : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">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"> ( not <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<b>thus </b><a NAME="E6:11"/>
not <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</font></i></a></i>; <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">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:11"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E9:11"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</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="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>hence </b><a NAME="E2:11_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E8:11"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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="c5">l</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E11:11"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="txt" href="#E8:11"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/>
<a NAME="E12:11"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:11"/>
not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self" title="PENCIL_1:def.4">Def4</a></i>; <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">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">F</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 <font color="Maroon" title="c1">X</font> : 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E12:11"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c7">L</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:11_2"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> &amp; 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">L</font> )
 ;<br/><b>thus </b><a NAME="E4:11_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:11_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</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>
<a NAME="E16:11"/><b>then </b>
<font color="Maroon" title="c1">X</font> is  not    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">F</font>
 
;<br/>
<b>hence </b><a NAME="E17:11"/>
not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="pencil_1.html#D5" target="_self" title="PENCIL_1:def.5">Def5</a></i>; <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">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<a NAME="E18:11"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1"><b>suppose </b></a><a NAME="E1:11_3_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><b>consider </b><font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_3_1_1"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E19">B2</font></i></a>, <a class="ref" href="pencil_1.html#T3" target="_self" title="PENCIL_1:th.3">Th3</a>, <a class="txt" href="#E1:11"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c9">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E3:11_3_1_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/><a NAME="E5:11_3_1_1"/><i><font color="Green" title="E27">A10</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</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="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:11_3_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<a NAME="E2:11_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E1:11_3_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><a NAME="E6:11_3_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</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="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:11_3_1_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:11_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</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="c11">l</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E8:11_3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="#E3:11_3_1_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/><a NAME="E9:11_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E10:11_3_1_1"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="#E5:11_3_1_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2"><b>suppose </b></a><a NAME="E1:11_3_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> </span><br/><div class="add"><a NAME="E2:11_3_1_2"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/><a NAME="E3:11_3_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</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="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:11_3_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:11_3_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</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="c9">l</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E5:11_3_1_2"/>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E2:11_3_1_2"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:11_3_1_2"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="pencil_1.html#R1" title="PENCIL_1:pred.1">are_collinear</a> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self" title="PENCIL_1:def.1">Def1</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E19:11"/>
not <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V7" title="PENCIL_1:attr.7">truly-partial</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D8" target="_self" title="PENCIL_1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><br/>

<b>thus </b><a NAME="E20:11"/>
<font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">PENCIL_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">k</font></span><br/>

<a NAME="E1:11_4"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E12:11"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">m</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:11_4"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:11_4"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="#E3:11_4"><i><font color="Green" title="E25">A12</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>thus </b><a NAME="E21:11"/>
<font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V6" title="PENCIL_1:attr.6">identifying_close_blocks</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c8">l</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7" target="_self" title="PENCIL_1:def.7">PENCIL_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font></span>)</span> implies <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> )</span><br/>

<a NAME="E1:11_5"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E12:11"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">m</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:11_5"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:11_5"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:11_5"/><i><font color="Green" title="E26">A14</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></i>;<br/>
<a NAME="E7:11_5"/><i><font color="Green" title="E27">A15</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font> )</span><br/>

<b>assume </b><a NAME="E1:11_5_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font></span><br/>

<b>hence </b><a NAME="E2:11_5_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font>
 <b>by </b><i><a class="txt" href="#E6:11_5"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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>
<a NAME="E8:11_5"/><i><font color="Green" title="E28">A16</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="#E6:11_5"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/>
<a NAME="E9:11_5"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2
 
<b>by </b><i><a class="txt" href="#E3:11_5"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">k1</font> = <font color="Maroon" title="c7">k</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E11:11_5"/>
<font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">k1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E12:11_5"/><b>then </b><i><font color="Green" title="E29">A17</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k1</font>
 
<b>by </b><i><a class="txt" href="#E3:11_5"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E7:11_5"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="#E8:11_5"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:11_5"/>
<font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E12:11"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">n</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E15:11_5"/><i><font color="Green" title="E30">A18</font></i>: 
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<a NAME="E16:11_5"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2
 
<b>by </b><i><a class="txt" href="#E15:11_5"><i><font color="Green" title="E30">A18</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">l1</font> = <font color="Maroon" title="c8">l</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E18:11_5"/>
<font color="Maroon" title="c7">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">l1</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>hence </b><a NAME="E19:11_5"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font>
 <b>by </b><i><a class="txt" href="#E7:11_5"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="#E8:11_5"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="#E12:11_5"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="#E15:11_5"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="E22:11"/>
<font color="Maroon" title="c2">S</font> is <a href="pencil_1.html#V8" title="PENCIL_1:attr.8">without_isolated_points</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D9" target="_self" title="PENCIL_1:def.9">PENCIL_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being  <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">l</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:11_6"/><i><font color="Green" title="E25">A19</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E19">B2</font></i></a>, <a class="ref" href="pencil_1.html#T3" target="_self" title="PENCIL_1:th.3">Th3</a>, <a class="txt" href="#E1:11"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:11_6"/><i><font color="Green" title="E26">A20</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="#E2:11_6"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/>
<a NAME="E4:11_6"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</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="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:11_6_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:11_6_1"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_6_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E2:11_6"><i><font color="Green" title="E25">A19</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="c9">l</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E6:11_6"/>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E3:11_6"><i><font color="Green" title="E26">A20</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c9">l</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">l</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">l</font></span><br/>

<b>thus </b><a NAME="E8:11_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">l</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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>


</div>
