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

<span class="kw">let </span><font color="Maroon" title="c2">X0</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">X0</font> is  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>  implies ( <font color="Maroon" title="c2">X0</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  &amp; <font color="Maroon" title="c2">X0</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1"/>
not <font color="Maroon" title="c2">X0</font> is  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  &amp; <font color="Maroon" title="c2">X0</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  )</span><br/>

<a NAME="E2:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> &amp; not <font color="Olive" title="b1">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#D3" target="_self" title="TEX_2:def.3">Def3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E11">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> implies <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><a NAME="E2:47_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> implies <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><a NAME="E2:47_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_2"/>
<font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:47_1_1"/>
( <font color="Maroon" title="c2">X0</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  &amp; <font color="Maroon" title="c2">X0</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="borsuk_1.html#D14" title="BORSUK_1:def.14">BORSUK_1:def 14</a>, <a class="ref" href="tsep_1.html#D1" title="TSEP_1:def.1">TSEP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
