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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>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a><br/>

<b>set </b><font color="Maroon">R</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>;<br/>
<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D16">REWRITE1:def 16</a>,<a class="ref" href="abcmiz_0.html#D1" target="_self">ABCMIZ_0:def 1</a><br/>

<b>assume </b><a NAME="E2:4_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 ;<br/>

<b>assume </b><a NAME="E3:4_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E6:4_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E7:4_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E8:4_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, , <a class="ref" href="relset_1.html#T19">RELSET_1:19</a></i>;<br/>
<b>hence </b><a NAME="E9:4_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">S</font> )
 <b>by </b><i/>;<br/>


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