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<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compact1.html#D6" target="_self" title="COMPACT1:def.6">COMPACT1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> st <font color="Olive" title="b1">U</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">B</font> being    <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c4">xx</font><span class="kw"> such that </span><br/><a NAME="E3:22_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c5">B</font> is  <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compact1.html#D3" target="_self" title="COMPACT1:def.3">Def3</a></span>;<br/>
<a NAME="E4:22_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E5:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 
;<br/>
<a NAME="E6:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/>
<a NAME="E7:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">xx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">K</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">xx</font><span class="kw"> such that </span><br/><a NAME="E9:22_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c6">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E10:22_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c6">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">KK</font> = <font color="Maroon" title="c6">K</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E12:22_1_1"/>
<font color="Maroon" title="c6">K</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="compact1.html#D1" target="_self" title="COMPACT1:def.1">Def1</a></span>;<br/>
<a NAME="E13:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">KK</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></span>;<br/>
<a NAME="E14:22_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<a NAME="E15:22_1_1"/>
<font color="Maroon" title="c4">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E16:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E17:22_1_1"/>
<font color="Maroon" title="c7">KK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E18:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">KK</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E16:22_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="topmetr.html#T6" title="TOPMETR:th.6">TOPMETR:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:22_1_1"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> st <font color="Olive" title="b1">U</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></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/>


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