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

<span class="kw">let </span><font color="Maroon" title="c1">TM</font> be   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</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"> for <font color="Olive" title="b1">A</font> being  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">A</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b2">U1</font>, <font color="Olive" title="b3">U2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">U2</font> holds <br/> ex <font color="Olive" title="b4">V1</font>, <font color="Olive" title="b5">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b4">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U1</font> &amp; <font color="Olive" title="b5">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">U2</font> &amp; <font color="Olive" title="b4">V1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b5">V2</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">V1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b5">V2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">U2</font> holds <br/> ex <font color="Olive" title="b3">V1</font>, <font color="Olive" title="b4">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b3">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Olive" title="b4">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U2</font> &amp; <font color="Olive" title="b3">V1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">V2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">V1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">V2</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">U2</font> holds <br/> ex <font color="Olive" title="b3">V1</font>, <font color="Olive" title="b4">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b3">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Olive" title="b4">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U2</font> &amp; <font color="Olive" title="b3">V1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">V2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">V1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">V2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c4">U2</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font> implies  ex <font color="Olive" title="b1">V1</font>, <font color="Olive" title="b2">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> &amp; <font color="Olive" title="b2">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U2</font> &amp; <font color="Olive" title="b1">V1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">V2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">V2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:33"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">V1</font>, <font color="Olive" title="b2">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Olive" title="b1">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> &amp; <font color="Olive" title="b2">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U2</font> &amp; <font color="Olive" title="b1">V1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">V2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">V2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">U12</font> = <font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">TU</font> = <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span>;<br/>
<a NAME="E3:33"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c8">u2</font> = <font color="Maroon" title="c4">U2</font>, <font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:33"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c9">a</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T21" title="TOPDIM_1:th.21">TOPDIM_1:21</a></span>;<br/>
<a NAME="E6:33"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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">consider </span><font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:33_1"/>
( not <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U1</font> &amp; not <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">U2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>
<a NAME="E7:33"/>
<font color="Maroon" title="c4">U2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E8:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c8">u2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</a></span>;<br/>
<a NAME="E9:33"/>
<font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E10:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c7">u1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</a></span>;<br/>
<a NAME="E11:33"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T21" title="TOPDIM_1:th.21">TOPDIM_1:21</a></span>;<br/>
<a NAME="E12:33"/>
(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">a</font> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="topdim_1.html#T17" title="TOPDIM_1:th.17">TOPDIM_1:17</a>, <a class="ref" href="metrizts.html#T9" title="METRIZTS:th.9">METRIZTS:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">L</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:33"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ,<font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E15:33"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="topdim_2.html#T11" target="_self" title="TOPDIM_2:th.11">Th11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c12">v2</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:33"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
( <font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">v1</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">v2</font> )
 <span class="kw">and </span><br/><a NAME="E18:33"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">v2</font>
 <span class="kw">and </span><br/><a NAME="E19:33"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">v2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="metrizts.html#D3" title="METRIZTS:def.3">METRIZTS:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">V1</font> = <font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c14">V2</font> = <font color="Maroon" title="c12">v2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">V1</font> = <font color="Maroon" title="c13">V1</font>, <font color="Maroon" title="c16">V2</font> = <font color="Maroon" title="c14">V2</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="tsep_1.html#T9" title="TSEP_1:th.9">TSEP_1:9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">V2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <br/>( <font color="Maroon" title="c16">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> &amp; <font color="Olive" title="b1">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U2</font> &amp; <font color="Maroon" title="c16">V2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">V2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">V2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">V1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> &amp; <font color="Maroon" title="c15">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U2</font> &amp; <font color="Maroon" title="c16">V2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">V1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">V1</font> )</span><br/>

<a NAME="E22:33"/>
( <font color="Maroon" title="c7">u1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">v2</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">v1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:33"/>
( <font color="Maroon" title="c16">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> &amp; <font color="Maroon" title="c15">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U2</font> &amp; <font color="Maroon" title="c16">V2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">V1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">V1</font></span><br/>

<a NAME="E24:33"/>
<font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E25:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="topdim_1.html#T19" title="TOPDIM_1:th.19">TOPDIM_1:19</a></span>;<br/>
<a NAME="E26:33"/><span class="kw">then </span>
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T5" title="TOPDIM_1:th.5">TOPDIM_1:5</a></span>;<br/>
<a NAME="E27:33"/><span class="kw">then </span>
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E28:33"/><span class="kw">then </span>
<font color="Maroon" title="c10">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">a</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="topdim_1.html#T6" title="TOPDIM_1:th.6">TOPDIM_1:6</a></span>;<br/>
<a NAME="E29:33"/><span class="kw">then </span>
<font color="Maroon" title="c10">L</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:33"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:33"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</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>
