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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"> for <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> holds  <font color="Maroon" title="c1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="interva1.html#K10" title="INTERVA1:func.10">^</a> </span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <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></span><br/>



<a NAME="E1:68"/><span class="lab"><font color="Green" title="E48">b1</font></span>: 
<font color="Maroon" title="c1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E49">c1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><br/>

<a NAME="E2:68_1"/>
<font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T16" target="_self" title="INTERVA1:th.16">wazne2</a></span>;<br/>
<a NAME="E3:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">c2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></span>;<br/>
<a NAME="E4:68_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E49">c1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:68_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </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="#E3:68_1"><span class="lab"><font color="Green" title="E50">c2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:68"/><span class="lab"><font color="Green" title="E49">a1</font></span>: 
<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#K6" title="INTERVA1:func.6">``2</a> </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"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </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="ref" href="interva1.html#T45" target="_self" title="INTERVA1:th.45">ThDop0</a></span>;<br/>
<a NAME="E3:68"/>
<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="E4:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">ab</font></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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<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="#E2:68"><span class="lab"><font color="Green" title="E49">a1</font></span></a></span>;<br/>
<a NAME="E5:68"/>
<font color="Maroon" title="c2">A</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <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="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#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></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#K6" title="INTERVA1:func.6">``2</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K5" title="INTERVA1:func.5">``1</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E50">ab</font></span></a>, <a class="ref" href="interva1.html#T17" target="_self" title="INTERVA1:th.17">Un</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:68"/>
<font color="Maroon" title="c1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E48">b1</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/>


</div>
