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<b>consider </b><font color="Maroon" title="c2">it0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">R</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">it0</font> iff ( <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">R</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:6_1_1"/>
for <font color="Olive" title="b1">R</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">it0</font> holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E2:6_1_1"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">it0</font> = <font color="Maroon" title="c2">it0</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">it0</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">R</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>thus </b><a NAME="E5:6_1_1"/>
( <font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it0</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">R</font>] )
 <b>by </b><i><a class="txt" href="arrow.html#E2:6_1_1"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>

<b>assume </b><a NAME="E6:6_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">R</font>]
 ;<br/>

<b>hence </b><a NAME="E7:6_1_1"/>
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it0</font>
 <b>by </b><i><a class="txt" href="arrow.html#E2:6_1_1"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>


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