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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<a NAME="E1:12"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:12"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:12"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:12"/><b>then </b>
<font color="Maroon">B</font> <a href="fin_topo.html#K12">^f</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:12"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K12">^f</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:12"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:12"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#T16">FIN_TOPO:16</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>suppose </b></a><a NAME="E1:12_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a> 
 <b>by </b><i>, <a class="ref" href="fin_topo.html#T16">FIN_TOPO:16</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K12">^f</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><b>suppose </b></a><a NAME="E1:12_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:12_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="fin_topo.html#K12">^f</a> 
 <b>by </b><i>, <a class="ref" href="fin_topo.html#T16">FIN_TOPO:16</a></i>;<br/><b>hence </b><a NAME="E3:12_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K12">^f</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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