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<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c4">s1</font> )
 ;<br/>
<a NAME="E4:20"/><i><font color="Green" title="E8">A3</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E1:20"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="triang_1.html#E3:20"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E5:20"/>
( <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">s1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E6:20"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E1:20"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="triang_1.html#E3:20"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E8:20"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> : the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="triang_1.html#E4:20"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:20"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>
 ;<br/>


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