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<b>let </b><font color="Maroon">R</font> be  <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">a</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="E1:26"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lattice3.html#K7">~</a> </span>)</span>
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:26"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:26"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lattice3.html#K7">~</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:26"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E10:26"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E11:26"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E12:26"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


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