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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:10"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:10"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<a NAME="E6:10"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E7:10"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E9:10"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></span>;<br/>
<a NAME="E10:10"/><span class="kw">then </span>
<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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E11:10"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="triang_1.html#K1" title="TRIANG_1:func.1">symplexes</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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