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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">WH</font> being   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  not <font color="Olive" title="b1">WH</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Olive" title="b1">WH</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">WH</font> be   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">WH</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Maroon" title="c2">WH</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:135"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
 not <font color="Maroon" title="c2">WH</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:135"/><span class="lab"><font color="Green" title="E96">A2</font></span>: 
<font color="Maroon" title="c2">WH</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span><br/>

<a NAME="E3:135"/><span class="lab"><font color="Green" title="E97">A3</font></span>: 
<font color="Maroon" title="c2">WH</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="135_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:135_1"/>
( <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T27" target="_self" onmouseover="rs('jordan2c/T27')" onmouseout="rh()">Th18</a>, <a class="ref" href="subset_1.html#T10" onmouseover="rs('subset_1/T10')" onmouseout="rh()">SUBSET_1:10</a></span>;<br/>
<a NAME="E2:135_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A4</font></span>: 
<font color="Maroon" title="c2">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="xboole_1.html#T73" onmouseover="rs('xboole_1/T73')" onmouseout="rh()">XBOOLE_1:73</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:135_1"/><span class="lab"><font color="Green" title="E99">A5</font></span>: 
<font color="Maroon" title="c2">WH</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:135_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T106" target="_self" onmouseover="rs('jordan2c/T106')" onmouseout="rh()">Th90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:135_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E3:135_1"><span class="lab"><font color="Green" title="E99">A5</font></span></a>, <a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E98">A4</font></span></a>, <a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</a>, <a class="ref" href="xboole_1.html#T73" onmouseover="rs('xboole_1/T73')" onmouseout="rh()">XBOOLE_1:73</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:135"/>
<font color="Maroon" title="c2">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="subset_1.html#T23" onmouseover="rs('subset_1/T23')" onmouseout="rh()">SUBSET_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:135"/>
<font color="Maroon" title="c2">WH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="ref" href="jordan2c.html#T124" target="_self" onmouseover="rs('jordan2c/T124')" onmouseout="rh()">Th108</a>, <a class="ref" href="goboard9.html#T4" onmouseover="rs('goboard9/T4')" onmouseout="rh()">GOBOARD9:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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