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<b>let </b><font color="Maroon" title="c3">P</font> be  <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a>  <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ;<br/><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:157"/><i><font color="Green" title="E59">a0</font></i>: 
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font>;<br/>
<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">p'</font>, <font color="Maroon" title="c8">q</font>, <font color="Maroon" title="c9">q'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">PCS_0:def 22</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:157"/><i><font color="Green" title="E60">r1</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">q</font>
 <b>and </b><br/><a NAME="E3:157"/><i><font color="Green" title="E61">r2</font></i>: 
<font color="Maroon" title="c7">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">p</font>
 <b>and </b><br/><a NAME="E4:157"/><i><font color="Green" title="E62">r3</font></i>: 
<font color="Maroon" title="c9">q'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:157_1"/>
( <font color="Maroon" title="c7">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or ( <font color="Maroon" title="c7">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c9">q'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_1"><b>suppose </b></a><a NAME="E1:157_1_1"/>
( <font color="Maroon" title="c7">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:157_1_1"/>
<font color="Maroon" title="c7">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="ref" href="pcs_0.html#T27" target="_self" title="PCS_0:th.27">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:157_1_2"/><i><font color="Green" title="E63">s1</font></i>: 
<font color="Maroon" title="c7">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E2:157_1_2"/><i><font color="Green" title="E64">s2</font></i>: 
<font color="Maroon" title="c9">q'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">p'0</font> = <font color="Maroon" title="c7">p'</font>, <font color="Maroon" title="c11">q'0</font> = <font color="Maroon" title="c9">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="pcs_0.html#E1:157_1_2"><i><font color="Green" title="E63">s1</font></i></a>, <a class="txt" href="pcs_0.html#E2:157_1_2"><i><font color="Green" title="E64">s2</font></i></a>, <a class="ref" href="pcs_0.html#T25" target="_self" title="PCS_0:th.25">Th10</a></i>;<br/><a NAME="E4:157_1_2"/><i><font color="Green" title="E65">f2</font></i>: 
( <font color="Maroon" title="c10">p'0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c11">q'0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:157_1_2"><i><font color="Green" title="E63">s1</font></i></a>, <a class="txt" href="pcs_0.html#E2:157_1_2"><i><font color="Green" title="E64">s2</font></i></a></i>;<br/><a NAME="E5:157_1_2"/><b>then </b><i><font color="Green" title="E66">f1</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:157"><i><font color="Green" title="E59">a0</font></i></a>, <a class="txt" href="pcs_0.html#E3:157"><i><font color="Green" title="E61">r2</font></i></a>, <a class="txt" href="pcs_0.html#E4:157"><i><font color="Green" title="E62">r3</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">p0</font> = <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c13">q0</font> = <font color="Maroon" title="c8">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="pcs_0.html#E4:157_1_2"><i><font color="Green" title="E65">f2</font></i></a>, <a class="txt" href="pcs_0.html#E1:157"><i><font color="Green" title="E59">a0</font></i></a>, <a class="txt" href="pcs_0.html#E3:157"><i><font color="Green" title="E61">r2</font></i></a>, <a class="txt" href="pcs_0.html#E4:157"><i><font color="Green" title="E62">r3</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><a NAME="E7:157_1_2"/><i><font color="Green" title="E67">f3</font></i>: 
<font color="Maroon" title="c12">p0</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c13">q0</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E5:157_1_2"><i><font color="Green" title="E66">f1</font></i></a>, <a class="txt" href="pcs_0.html#E2:157"><i><font color="Green" title="E60">r1</font></i></a>, <a class="ref" href="pcs_0.html#T29" target="_self" title="PCS_0:th.29">Th12a</a></i>;<br/><a NAME="E8:157_1_2"/>
( <font color="Maroon" title="c10">p'0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">p0</font> &amp; <font color="Maroon" title="c11">q'0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">q0</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:157"><i><font color="Green" title="E61">r2</font></i></a>, <a class="txt" href="pcs_0.html#E4:157"><i><font color="Green" title="E62">r3</font></i></a>, <a class="txt" href="pcs_0.html#E1:157_1_2"><i><font color="Green" title="E63">s1</font></i></a>, <a class="txt" href="pcs_0.html#E2:157_1_2"><i><font color="Green" title="E64">s2</font></i></a>, <a class="ref" href="pcs_0.html#T26" target="_self" title="PCS_0:th.26">Th12</a></i>;<br/><a NAME="E9:157_1_2"/><b>then </b>
<font color="Maroon" title="c10">p'0</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c11">q'0</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E7:157_1_2"><i><font color="Green" title="E67">f3</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def4</a></i>;<br/><b>hence </b><a NAME="E10:157_1_2"/>
<font color="Maroon" title="c7">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="ref" href="pcs_0.html#T28" target="_self" title="PCS_0:th.28">Th15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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