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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b1">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> holds <br/> <a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Olive" title="b1">a</font> is  <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> implies  <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> is  <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:156"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> is  <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a> </span><br/>

<span class="kw">set </span><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/>
<span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">p9</font>, <font color="Maroon" title="c8">q</font>, <font color="Maroon" title="c9">q9</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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">PCS_0:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c7">p9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c9">q9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font> implies <font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:156"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:156"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E4:156"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c9">q9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:156_1"/>
( <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or ( <font color="Maroon" title="c7">p9</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">q9</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="156_1_1"><span class="kw">suppose </span></a><a NAME="E1:156_1_1"/>
( <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:156_1_1"/>
<font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#T27" target="_self" title="PCS_0:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:156_1_2"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E2:156_1_2"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c9">q9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c10">p90</font> = <font color="Maroon" title="c7">p9</font>, <font color="Maroon" title="c11">q90</font> = <font color="Maroon" title="c9">q9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_1_2"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E2:156_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="pcs_0.html#T25" target="_self" title="PCS_0:th.25">Th25</a></span>;<br/><a NAME="E4:156_1_2"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c10">p90</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_1_2"><span class="lab"><font color="Green" title="E63">A5</font></span></a></span>;<br/><a NAME="E5:156_1_2"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c11">q90</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a></span>;<br/><a NAME="E6:156_1_2"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E4:156_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th24</a></span>;<br/><a NAME="E7:156_1_2"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:156"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E5:156_1_2"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th24</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E4:156"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E4:156_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E5:156_1_2"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th24</a></span>;<br/><a NAME="E9:156_1_2"/><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E6:156_1_2"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E7:156_1_2"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="pcs_0.html#T29" target="_self" title="PCS_0:th.29">Th29</a></span>;<br/><a NAME="E10:156_1_2"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
<font color="Maroon" title="c10">p90</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">p0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E1:156_1_2"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="pcs_0.html#T26" target="_self" title="PCS_0:th.26">Th26</a></span>;<br/><a NAME="E11:156_1_2"/>
<font color="Maroon" title="c11">q90</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">q0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E2:156_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="pcs_0.html#T26" target="_self" title="PCS_0:th.26">Th26</a></span>;<br/><a NAME="E12:156_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p90</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c11">q90</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:156_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E10:156_1_2"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></span>;<br/><span class="kw">hence </span><a NAME="E13:156_1_2"/>
<font color="Maroon" title="c7">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c9">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#T28" target="_self" title="PCS_0:th.28">Th28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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