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<a NAME="E1:97_1_1"/>
 <a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font> is <a href="borsuk_1.html#V2" title="BORSUK_1:attr.2">DECOMPOSITION-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:97_1_1_1"/><i><font color="Green" title="E72">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D15" target="_self" title="BORSUK_1:def.15">BORSUK_1:def 15</a><br/>

<b>assume </b><a NAME="E2:97_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">W</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E4:97_1_1_1"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T32" target="_self" title="BORSUK_1:th.32">Th32</a></i>;<br/>
<a NAME="E5:97_1_1_1"/><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:97_1_1_1"><i><font color="Green" title="E73">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_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="E1:97_1_1_1_1_1"/>
( <font color="Maroon" title="c5">W</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">X</font> or  not <font color="Maroon" title="c5">W</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">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:97_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">W</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">X</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">W'</font> = <font color="Maroon" title="c5">W</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X</font> ;<br/><a NAME="E3:97_1_1_1_1_1_1"/><i><font color="Green" title="E75">A4</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">W'</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:97_1_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T76" target="_self" title="BORSUK_1:th.76">Th76</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="borsuk_1.html#E1:97_1_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:97_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">B</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:97_1_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E3:97_1_1_1_1_1_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="borsuk_1.html#D15" target="_self" title="BORSUK_1:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E6:97_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T42" target="_self" title="BORSUK_1:th.42">Th42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:97_1_1_1_1_1_2"/>
not <font color="Maroon" title="c5">W</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">X</font>
 ;<br/><div class="add"><a NAME="E2:97_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:97_1_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T77" target="_self" title="BORSUK_1:th.77">Th77</a></i>;<br/><b>hence </b><a NAME="E3:97_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T41" target="_self" title="BORSUK_1:th.41">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:97_1_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:97_1_1"/>
 <a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font> is   <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Maroon" title="c1">XX</font>
 ;<br/>


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