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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:70"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#T20">WELLORD1:20</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>thus </b><a NAME="E2:70"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:70_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:70_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:70_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:70_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:70_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:70_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E6:70_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:70_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/>


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


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