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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="orders_1.html#M1">Choice_Function</a> of  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">fC1</font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font>;<br/><b>let </b><font color="Maroon">fC2</font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font>;<br/>







<b>thus </b><a NAME="E1:62"/>
( <font color="Maroon">fC1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">fC2</font> implies <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">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">E18</font></i>: 
<font color="Maroon">fC1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">fC2</font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">fC1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fC2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">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">E39</font></i>: 
<font color="Maroon">fC2</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font>
 ;<br/><a NAME="E2:62_1_1"/>
<font color="Maroon">fC1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E3:62_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> &amp; <font color="Maroon">fC2</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:62_1_1"/><b>then </b>
<font color="Maroon">fC2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fC1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:62_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1"/>
<font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font>
 ;<br/>

<a NAME="E3:62"/>
<font color="Maroon">fC2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E4:62"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font> &amp; <font color="Maroon">fC1</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:62"/><b>then </b>
( <font color="Maroon">fC1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">fC2</font> &amp; <font color="Maroon">fC1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fC2</font> )
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#T24" target="_self">Th24</a></i>;<br/>
<b>hence </b><a NAME="E6:62"/>
<font color="Maroon">fC1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">fC2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>


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