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<a NAME="E1:100_1_1"/>
 <a href="borsuk_1.html#K17" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> is <a href="borsuk_1.html#V2" target="_self">DECOMPOSITION-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:100_1_1_1"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D15" target="_self">BORSUK_1:def 15</a><br/>

<b>assume </b><a NAME="E2:100_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K17" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:100_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2" target="_self">proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E5:100_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_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="E1:100_1_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E3:100_1_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:100_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#T76" target="_self">Th76</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:100_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="borsuk_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E6:100_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T42" target="_self">Th42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_1_1_2"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:100_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#T77" target="_self">Th77</a></i>;<br/><b>hence </b><a NAME="E3:100_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T41" target="_self">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:100_1_1"/>
 <a href="borsuk_1.html#K16" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="borsuk_1.html#NM2">DECOMPOSITION</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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