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<b>let </b><font color="Maroon">S</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">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">D2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font>;<br/>
<b>thus </b><a NAME="E1:47"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K3">"/\"</a> <font color="Maroon">D2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:47_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6">:]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6">:]</a></span> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:47_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel_2.html#E4:47_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i>, <a class="txt" href="waybel_2.html#E4:47_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:47_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:47_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:47_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K3">"/\"</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D4">YELLOW_4:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>reconsider </b><font color="Maroon">X</font> = <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6">:]</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>let </b><font color="Maroon">q</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="E3:47"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K3">"/\"</a> <font color="Maroon">D2</font>
 ;<br/>

<a NAME="E4:47"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D4">YELLOW_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 ;<br/>
<a NAME="E7:47"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:47"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E9:47"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


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