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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E3:5"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">V</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c1">L</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27" title="WAYBEL_0:def.27">WAYBEL_0:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E5:5"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c6">Z</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E6:5"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">Z</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c6">Z</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">Y</font> <b>by </b><i><a class="txt" href="waybel23.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Z1</font> = <font color="Maroon" title="c7">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:5"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c8">Z1</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E6:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c8">Z1</font>,<font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c8">Z1</font>,<font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="waybel23.html#E6:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_0.html#T65" title="YELLOW_0:th.65">YELLOW_0:65</a></i>;<br/>
<a NAME="E11:5"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c2">S</font></span>)</span> where <font color="Olive" title="b1">V</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">Y</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c2">S</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27" title="WAYBEL_0:def.27">WAYBEL_0:def 27</a></i>;<br/>


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