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<div class="add">

<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="tsp_1.html#E29"><i><font color="Green" title="E22">Lm3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:36_1_1"/>
(  not <font color="Maroon" title="c2">Y0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a> or <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or 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="c1">Y</font>  st <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b2">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b3">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b3">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">Y0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_1_1_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
not <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>assume </b><a NAME="E2:36_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Maroon" title="c5">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">x0</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y0</font> = <font color="Maroon" title="c5">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> ;<br/><a NAME="E4:36_1_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:36_1_1_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>assume </b><a NAME="E5:36_1_1_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_1_1_1"/>
(  ex <font color="Olive" title="b1">E0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> &amp; not <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> ) or  ex <font color="Olive" title="b1">F0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> ) )
 </span><b>by </b><i><a class="txt" href="tsp_1.html#E1:36_1_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="tsp_1.html#E1:36_1_1_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="tsp_1.html#E5:36_1_1_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="tsp_1.html#D4" target="_self" title="TSP_1:def.4">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">E0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> &amp; not <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">E0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font><b> such that </b><br/><a NAME="E3:36_1_1_1_1_1_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c8">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:36_1_1_1_1_1_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E0</font>
 <b>and </b><br/><a NAME="E5:36_1_1_1_1_1_1_1"/><i><font color="Green" title="E29">A7</font></i>: 
not <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E2:36_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c9">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E3:36_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c8">E0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="tsp_1.html#D2" target="_self" title="TSP_1:def.2">Def2</a></i>;<br/><b>take </b><font color="Maroon" title="c10">E</font> = <font color="Maroon" title="c9">E</font>;<br/><b>thus </b><a NAME="E4:36_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_1_1_1_1_1_1"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><a NAME="E5:36_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E</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="E6:36_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:36_1_1_1_1_1_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_1_1"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E7:36_1_1_1_1_1_1_1_1"/>
not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:36_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="tsp_1.html#E5:36_1_1_1_1_1_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_1_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_1_1_1_1"/>
(  ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) or  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">F0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">F0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font><b> such that </b><br/><a NAME="E3:36_1_1_1_1_1_1_2"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c8">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:36_1_1_1_1_1_1_2"/><i><font color="Green" title="E28">A11</font></i>: 
not <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F0</font>
 <b>and </b><br/><a NAME="E5:36_1_1_1_1_1_1_2"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c7">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E2:36_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c9">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E3:36_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c8">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="tsp_1.html#D2" target="_self" title="TSP_1:def.2">Def2</a></i>;<br/><b>take </b><font color="Maroon" title="c10">F</font> = <font color="Maroon" title="c9">F</font>;<br/><b>thus </b><a NAME="E4:36_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_1_1_1_1_2_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/><b>thus </b><a NAME="E5:36_1_1_1_1_1_1_2_1"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:36_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="tsp_1.html#E4:36_1_1_1_1_1_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_2_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:36_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F</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="E7:36_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E5:36_1_1_1_1_1_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_1_1_1_1_2_1"><i><font color="Green" title="E31">A14</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_1_1_1_2"/>
(  ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) or  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_1"/>
(  ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) or  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )
 ;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:36_1_1"/><i><font color="Green" title="E23">A15</font></i>: 
( <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or 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="c1">Y</font>  st <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b2">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b3">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b3">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_2"/><i><font color="Green" title="E24">A16</font></i>: 
not <font color="Maroon" title="c2">Y0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><b>let </b><font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c5">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font>;<br/><a NAME="E2:36_1_1_2"/><i><font color="Green" title="E25">A17</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:36_1_1_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:36_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:36_1_1_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:36_1_1_2"/><b>then </b><i><font color="Green" title="E26">A18</font></i>: 
( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> )
 ;<br/><a NAME="E5:36_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#D1" target="_self" title="TSP_1:def.1">Def1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="tsp_1.html#E4:36_1_1_2"><i><font color="Green" title="E26">A18</font></i></a></i>;<br/><b>assume </b><a NAME="E7:36_1_1_2"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2_1_1"/>
(  ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) or  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )
 </span><b>by </b><i><a class="txt" href="tsp_1.html#E3:36_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="tsp_1.html#E1:36_1_1_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="tsp_1.html#E7:36_1_1_2"><i><font color="Green" title="E27">A19</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:36_1_1_2_1_1_1"/><i><font color="Green" title="E28">A20</font></i>: 
<font color="Maroon" title="c8">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:36_1_1_2_1_1_1"/><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 <b>and </b><br/><a NAME="E5:36_1_1_2_1_1_1"/><i><font color="Green" title="E30">A22</font></i>: 
not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">E0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font><b> such that </b><br/><a NAME="E2:36_1_1_2_1_1_1_1"/><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c9">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E3:36_1_1_2_1_1_1_1"/><i><font color="Green" title="E32">A24</font></i>: 
<font color="Maroon" title="c9">E0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="tsp_1.html#T4" target="_self" title="TSP_1:th.4">Th4</a></i>;<br/><b>take </b><font color="Maroon" title="c10">E0</font> = <font color="Maroon" title="c9">E0</font>;<br/><b>thus </b><a NAME="E4:36_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_2_1_1_1_1"><i><font color="Green" title="E31">A23</font></i></a></i>;<br/><b>thus </b><a NAME="E5:36_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E0</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_2"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="tsp_1.html#E4:36_1_1_2_1_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_1_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E0</font>
 ;<br/><a NAME="E2:36_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</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="E3:36_1_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E5:36_1_1_2_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_1_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="tsp_1.html#E1:36_1_1_2_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_2_1_1_1_1"/>
not <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E0</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_2_1_1_1"/>
(  ex <font color="Olive" title="b1">E0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> &amp; not <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> ) or  ex <font color="Olive" title="b1">F0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:36_1_1_2_1_1_2"/><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c8">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:36_1_1_2_1_1_2"/><i><font color="Green" title="E29">A27</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 <b>and </b><br/><a NAME="E5:36_1_1_2_1_1_2"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">F0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font><b> such that </b><br/><a NAME="E2:36_1_1_2_1_1_2_1"/><i><font color="Green" title="E31">A29</font></i>: 
<font color="Maroon" title="c9">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E3:36_1_1_2_1_1_2_1"/><i><font color="Green" title="E32">A30</font></i>: 
<font color="Maroon" title="c9">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="tsp_1.html#T4" target="_self" title="TSP_1:th.4">Th4</a></i>;<br/><b>take </b><font color="Maroon" title="c10">F0</font> = <font color="Maroon" title="c9">F0</font>;<br/><b>thus </b><a NAME="E4:36_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c10">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_2_1_1_2_1"><i><font color="Green" title="E31">A29</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E33">A31</font></i>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F0</font>
 ;<br/><a NAME="E2:36_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c8">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</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="E3:36_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:36_1_1_2_1_1_2"><i><font color="Green" title="E29">A27</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_2_1"><i><font color="Green" title="E32">A30</font></i></a>, <a class="txt" href="tsp_1.html#E1:36_1_1_2_1_1_2_1_1"><i><font color="Green" title="E33">A31</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:36_1_1_2_1_1_2_1"/>
not <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F0</font>
 ;<br/><b>thus </b><a NAME="E7:36_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F0</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:36_1_1_2"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="tsp_1.html#E5:36_1_1_2_1_1_2"><i><font color="Green" title="E30">A28</font></i></a>, <a class="txt" href="tsp_1.html#E3:36_1_1_2_1_1_2_1"><i><font color="Green" title="E32">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_2_1_1_2"/>
(  ex <font color="Olive" title="b1">E0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> &amp; not <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> ) or  ex <font color="Olive" title="b1">F0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:36_1_1_2"/>
(  ex <font color="Olive" title="b1">E0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">E0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> &amp; not <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E0</font> ) or  ex <font color="Olive" title="b1">F0</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y0</font> st <br/>( <font color="Olive" title="b1">F0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Maroon" title="c5">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F0</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_1_1"/>
<font color="Maroon" title="c2">Y0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D4" target="_self" title="TSP_1:def.4">Def4</a></i>;<br/>


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