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<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>, <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">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:36"/><i><font color="Green">E27</font></i>: 
for <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 <font color="Olive">b<sub>1</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="Olive">b<sub>2</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> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_2.html#D2" target="_self">TSEP_2:def 2</a><br/>

<b>thus </b><a NAME="E3:36"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tsep_1.html#V1">open</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="borsuk_1.html#V1">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</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> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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="E1:36_1_1"/>
<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>3</sub></font>
 ;<br/><a NAME="E2:36_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="tsep_2.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tsep_2.html#E1:36_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="borsuk_1.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a></i>;<br/>


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

<b>assume </b><a NAME="E4:36"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</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> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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="E1:36_2"/>
<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>2</sub></font>
 ;<br/><a NAME="E2:36_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a>  &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_2.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tsep_2.html#E4:36"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tsep_2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tsep_1.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a></i>;<br/>


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