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<b>let </b><font color="Maroon" title="c1">A</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <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/> for <font color="Olive" title="b2">fC1</font>, <font color="Olive" title="b3">fC2</font> being   <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Olive" title="b1">f</font> holds <br/> ( <font color="Olive" title="b2">fC1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">fC2</font> iff <font color="Olive" title="b2">fC1</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Olive" title="b3">fC2</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">fC1</font>, <font color="Olive" title="b2">fC2</font> being   <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c2">f</font> holds <br/> ( <font color="Olive" title="b1">fC1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">fC2</font> iff <font color="Olive" title="b1">fC1</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Olive" title="b2">fC2</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> iff <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> )</span><br/>







<b>thus </b><a NAME="E1:68"/>
( <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E42">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1_1"/><i><font color="Green" title="E43">A2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:68_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="E3:68_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="#E1:68_1_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/><a NAME="E4:68_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="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E5:68_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:68_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_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="#E1:68_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self" title="ORDERS_2:th.83">Th83</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:68"/><i><font color="Green" title="E42">A3</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<a NAME="E3:68"/>
<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="E4:68"/><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="#E2:68"><i><font color="Green" title="E42">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="E5:68"/><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="#E2:68"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="orders_2.html#T68" target="_self" title="ORDERS_2:th.68">Th68</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:68"/>
<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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