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<b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be   <a href="pcs_0.html#NM1" title="PCS_0:NM.1">pcs</a>;<br/>

<b>assume </b><a NAME="E1:141"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">Q</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>;<br/>
<a NAME="E2:141"/>
 <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="pcs_0.html#V14" title="PCS_0:attr.14">pcs-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>proof </b></a><div class="add">

<a NAME="E1:141_1"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<b>thus </b><a NAME="E2:141_1"/>
 <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D23" target="_self" title="PCS_0:def.23">PCS_0:def 23</a><br/>

<b>thus </b><a NAME="E3:141_1"/>
 <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:141_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:141"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="pcs_0.html#T18" target="_self" title="PCS_0:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E2:141_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:141_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a><br/>


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

<b>thus </b><a NAME="E4:141_1"/>
(  <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a> &amp;  <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a> )
 ;<br/>

<b>thus </b><a NAME="E5:141_1"/>
 <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:141"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="pcs_0.html#T19" target="_self" title="PCS_0:th.19">Th19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:141"/>
 <a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> is   <a href="pcs_0.html#NM1" title="PCS_0:NM.1">pcs</a>
 ;<br/>


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