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<b>consider </b><font color="Maroon">R</font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>
<b>set </b><font color="Maroon">J</font> = <font color="Maroon">I</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R</font>;<br/>
<a NAME="E2:47_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">R</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">I</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>thus </b><a NAME="E4:47_1_1"/>
<font color="Maroon">J</font> is <a href="yellow_1.html#V1">RelStr-yielding</a>
 ;<br/>

<b>thus </b><a NAME="E5:47_1_1"/>
<font color="Maroon">J</font> is <a href="waybel_3.html#V4" target="_self">non-Empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be    <a href="struct_0.html#L1">1-sorted</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D7" target="_self">WAYBEL_3:def 7</a><br/>

<b>assume </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">J</font>
 ;<br/>

<b>hence </b><a NAME="E2:47_1_1_1"/>
not <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">S</font> be    <a href="orders_2.html#L1">RelStr</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D8" target="_self">WAYBEL_3:def 8</a><br/>

<b>assume </b><a NAME="E6:47_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">J</font>
 ;<br/>

<b>hence </b><a NAME="E7:47_1_1"/>
<font color="Maroon">S</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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