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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font> iff for <font color="Olive" title="b2">O</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> holds <br/> not <font color="Olive" title="b2">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>let </b><font color="Maroon" title="c2">r3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font> iff for <font color="Olive" title="b1">O</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/> not <font color="Olive" title="b1">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">O</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/> not <font color="Olive" title="b1">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) implies <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">O</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/> not <font color="Olive" title="b1">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>let </b><font color="Maroon" title="c3">O</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> implies  not <font color="Maroon" title="c3">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E2:55_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font>
 <b>and </b><br/><a NAME="E3:55_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:55_1"/>
<font color="Maroon" title="c3">O</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:55_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:55_1"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">O</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E6:55_1"/>
<font color="Maroon" title="c3">O</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></i>;<br/><a NAME="E7:55_1"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">O</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E5:55_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="pscomp_1.html#T32" target="_self" title="PSCOMP_1:th.32">Th32</a></i>;<br/><a NAME="E8:55_1"/>
<font color="Maroon" title="c3">O</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">O</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/><b>hence </b><a NAME="E9:55_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:55_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E2:55_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E7:55_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E34">A6</font></i>: 
for <font color="Olive" title="b1">O</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/> not <font color="Olive" title="b1">O</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E3:55"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T33" target="_self" title="PSCOMP_1:th.33">Th33</a></i>;<br/>
<a NAME="E6:55"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:55"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="#E5:55"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:55"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="subset_1.html#T40" title="SUBSET_1:th.40">SUBSET_1:40</a></i>;<br/>
<a NAME="E9:55"/><b>then </b>
not <font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="#E2:55"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E4:55"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:55"/>
<font color="Maroon" title="c2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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