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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">o</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">o'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E17">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Olive" title="b2">b</font>
 ;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a</font> )
 </span><b>by </b><i><a class="ref" href="arrow.html#T6" target="_self" title="ARROW:th.6">Th6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> iff <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E1:22_1"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_2"/>
( <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> iff <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="ref" href="arrow.html#T3" target="_self" title="ARROW:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_1_3"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="arrow.html#E1:22_1"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:22_1_1_3"/>
( <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> iff <font color="Maroon" title="c4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E1:22_1_1_3"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:22"/>
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">b</font> iff <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Olive" title="b2">b</font> ) ) implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">o'</font>,<font color="Olive" title="b2">b</font> )
 ;<br/>


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