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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>thus </b><a NAME="E3:91"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a>  implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:91_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E5:91_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E6:91_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E7:91_1"/><i><font color="Green">E60</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E4:91_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:91_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:91_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tsep_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:91_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:91_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="tsep_1.html#E7:91_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tsep_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E12:91_1"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:91_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="tsep_1.html#E5:91_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="tsep_1.html#E9:91_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="tsep_1.html#E11:91_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font><b> such that </b><a NAME="E4:91"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E5:91"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:91_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:91_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E3:91_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:91"><i><font color="Green">E57</font></i></a>, <a class="txt" href="tsep_1.html#E5:91"><i><font color="Green">E58</font></i></a>, <a class="txt" href="tsep_1.html#E1:91_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, <a class="ref" href="tsep_1.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:91_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:91"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
