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<span class="kw">let </span><font color="Maroon" title="c1">U</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> st <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span></span><br/>

<a NAME="E1:65"/><span class="lab"><font color="Green" title="E48">a0</font></span>: 
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">A</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font><span class="kw"> such that </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E49">a1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span>
 ;<br/>
<span class="lab"><font color="Green" title="E50">d2</font></span>: <font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a>  = 
 <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">a1</font></span></a>, <a class="ref" href="interva1.html#T46" target="_self" title="INTERVA1:th.46">ThDop1</a></span>
<br/>.= 
 <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span>

;<br/>
<a NAME="E5:65"/>
<font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a> </span>)</span> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a> </span>)</span> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T15" target="_self" title="INTERVA1:th.15">Wazne33</a></span>;<br/>
<a NAME="E6:65"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a> </span>)</span> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a> </span>)</span> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T6" target="_self" title="INTERVA1:th.6">Pik</a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E50">d2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">a2</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a>  = 
 <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">a1</font></span></a>, <a class="ref" href="interva1.html#T18" target="_self" title="INTERVA1:th.18">Int</a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E50">d2</font></span></a></span>
<br/>.= 
 <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font></span>)</span>

;<br/>
<a NAME="E8:65"/>
not  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font>
 
;<br/>
<a NAME="E9:65"/><span class="kw">then </span>
not  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T1" target="_self" title="INTERVA1:th.1">Lemacik</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:65"/>
 ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> st <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="interva1.html#K2" title="INTERVA1:func.2">Inter</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">U</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E51">a2</font></span></a>, <a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">a0</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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