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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  implies for <font color="Olive" title="b1">B</font> being  <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>  holds not <font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>  holds not <font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:40"/>
<font color="Maroon" title="c2">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">B1</font> = <font color="Maroon" title="c2">B</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:40"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow15.html#T19" target="_self" title="YELLOW15:th.19">Th19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
not <font color="Maroon" title="c4">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c4">X</font> as   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:40"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:40"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E13:40"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="yellow15.html#T4" target="_self" title="YELLOW15:th.4">Th4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E16:40"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c10">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E17:40"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
( ( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> ) or ( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="t_0topsp.html#D7" title="T_0TOPSP:def.7">T_0TOPSP:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1"/>
( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> ) or ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E24">A8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:40_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:40_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><a NAME="E4:40_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="yellow15.html#T30" target="_self" title="YELLOW15:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="40_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:40_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:40_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><a NAME="E4:40_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="yellow15.html#T30" target="_self" title="YELLOW15:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E1:40_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E19:40"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
