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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font>;<br/>





<b>set </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c1">X</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <font color="Maroon" title="c3">y</font>;<br/>
<b>set </b><font color="Maroon" title="c5">H</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c6">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E1:28"/>
<font color="Maroon" title="c6">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P1</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>
 <b>by </b><i><a class="ref" href="funcop_1.html#T20" title="FUNCOP_1:th.20">FUNCOP_1:20</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><div class="add"><a NAME="E2:28_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T22" title="FUNCOP_1:th.22">FUNCOP_1:22</a></i>;<br/><b>hence </b><a NAME="E3:28_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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