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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="latsum_1.html#K1" title="LATSUM_1:func.1">[*]</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="latsum_1.html#R1" title="LATSUM_1:pred.1">tolerates</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c2">S</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 ;<br/><a NAME="E2:13_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="latsum_1.html#K1" title="LATSUM_1:func.1">[*]</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:13_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="latsum_1.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="latsum_1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="latsum_1.html#E2:13_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="latsum_1.html#T5" target="_self" title="LATSUM_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E4:13_1"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:13"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">d</font>
 ;<br/>

<a NAME="E5:13"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E6:13"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="latsum_1.html#K1" title="LATSUM_1:func.1">[*]</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="latsum_1.html#E5:13"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="latsum_1.html#T6" target="_self" title="LATSUM_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E7:13"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


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