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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:106"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:106"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>assume </b><a NAME="E5:106"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_1"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 ;<br/><a NAME="E2:106_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_1_1"/>
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><a NAME="E2:106_1_1"/><b>then </b>
( not <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; not <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">C1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:106_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E5:106_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or  ex <font color="Olive">C</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T30" target="_self">Th30</a>, <a class="ref" href="tsep_1.html#T11" target="_self">Th11</a></i>;<br/><b>thus </b><a NAME="E6:106_1_1"/>
 ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:106_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, </i>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:106_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y1</font>
;<br/>

<b>take </b>
<font color="Maroon">Y2</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:106_1_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E4:106_1_1_1_1"/>
 ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:106_1_1_1_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<i><font color="Green">E95</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
;<br/>
<a NAME="E4:106_1_1_1_1_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/>
<b>hence </b><a NAME="E5:106_1_1_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:106_1_1_1"/>
( <font color="Maroon">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, <a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:106_1"/>
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_2_1"/>
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:106_2_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 <b>and </b><br/><a NAME="E4:106_2_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#T30" target="_self">Th30</a></i>;<br/><b>reconsider </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">C2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E7:106_2_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i/>;<br/><a NAME="E8:106_2_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_2_1_1_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:106_2_1_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:106_2_1_1_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E1:106_2_1_1_1_1_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:106_2_1_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:106_2_1_1_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:106_2_1_1_1_2_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:106_2_1_1_1_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:106"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E5:106_2_1_1_1_2_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, </i>;<br/>
<a NAME="E6:106_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, </i>;<br/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E8:106_2_1_1_1_2_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:106_2_1"/><b>then </b>
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> holds <br/><font color="Olive">A1</font>,<font color="Olive">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, </i>;<br/><b>hence </b><a NAME="E11:106_2_1"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:106_2"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:106"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  iff ( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) ) )
 <b>by </b><i/>;<br/>


</div>
