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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">fC1</font>, <font color="Maroon" title="c4">fC2</font> be    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c2">f</font>;<br/>







<b>thus </b><a NAME="E1:62"/>
( <font color="Maroon" title="c3">fC1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">fC2</font> implies <font color="Maroon" title="c3">fC1</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c4">fC2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c3">fC1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">fC2</font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">fC1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fC2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c4">fC2</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c3">fC1</font>
 ;<br/><a NAME="E2:62_1_1"/>
<font color="Maroon" title="c3">fC1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a></i>;<br/><a NAME="E3:62_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">fC1</font> &amp; <font color="Maroon" title="c4">fC2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">fC1</font>,<font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E1:62_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/><a NAME="E4:62_1_1"/><b>then </b>
<font color="Maroon" title="c4">fC2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">fC1</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E78"><i><font color="Green" title="E28">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:62_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E1:62_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="orders_2.html#E2:62_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1"/>
<font color="Maroon" title="c3">fC1</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c4">fC2</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:62_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self" title="ORDERS_2:th.83">Th83</a></i>;<br/>


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

<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c3">fC1</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c4">fC2</font>
 ;<br/>

<a NAME="E3:62"/>
<font color="Maroon" title="c4">fC2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a></i>;<br/>
<a NAME="E4:62"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fC2</font> &amp; <font color="Maroon" title="c3">fC1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c4">fC2</font>,<font color="Olive" title="b1">a</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:62"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
( <font color="Maroon" title="c3">fC1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">fC2</font> &amp; <font color="Maroon" title="c3">fC1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fC2</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:62"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="orders_2.html#E78"><i><font color="Green" title="E28">Lm3</font></i></a>, <a class="ref" href="orders_2.html#T68" target="_self" title="ORDERS_2:th.68">Th68</a></i>;<br/>
<b>hence </b><a NAME="E6:62"/>
<font color="Maroon" title="c3">fC1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">fC2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>


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