<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c3">P</font> be    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ;<br/><b>let </b><font color="Maroon" title="c4">D</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">P</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>thus </b><a NAME="E1:201"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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>  <a href="pcs_0.html#K24" title="PCS_0:func.24">pcs-general-power-IR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font> implies ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:201_1"/><i><font color="Green" title="E68">b0</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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>  <a href="pcs_0.html#K24" title="PCS_0:func.24">pcs-general-power-IR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font>
 ;<br/>

<b>hence </b><a NAME="E2:201_1"/><i><font color="Green" title="E69">b2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#D45" target="_self" title="PCS_0:def.45">Defp5</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>;<br/>

<b>assume </b><a NAME="E3:201_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:201_1"/><i><font color="Green" title="E70">b1</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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 <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:201_1"><i><font color="Green" title="E68">b0</font></i></a>, <a class="ref" href="pcs_0.html#D45" target="_self" title="PCS_0:def.45">Defp5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</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#E5:201_1"><i><font color="Green" title="E70">b1</font></i></a>, <a class="txt" href="pcs_0.html#E2:201_1"><i><font color="Green" title="E69">b2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">b</font>
;<br/>

<b>thus </b><a NAME="E7:201_1"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E5:201_1"><i><font color="Green" title="E70">b1</font></i></a>, <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><b>that </b><br/><a NAME="E2:201"/><i><font color="Green" title="E68">a1</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )
 <b>and </b><br/><a NAME="E3:201"/><i><font color="Green" title="E69">a2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> )
 ;<br/>

<a NAME="E4:201"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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 <font color="Maroon" title="c3">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:201_2"/><i><font color="Green" title="E70">a3</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</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#E2:201"><i><font color="Green" title="E68">a1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E4:201_2"/><i><font color="Green" title="E71">a4</font></i>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:201"><i><font color="Green" title="E69">a2</font></i></a>, <a class="txt" href="pcs_0.html#E1:201_2"><i><font color="Green" title="E70">a3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">b</font>
;<br/>

<b>thus </b><a NAME="E5:201_2"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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 <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:201_2"><i><font color="Green" title="E71">a4</font></i></a>, <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>hence </b><a NAME="E5:201"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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>  <a href="pcs_0.html#K24" title="PCS_0:func.24">pcs-general-power-IR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:201"><i><font color="Green" title="E68">a1</font></i></a>, <a class="ref" href="pcs_0.html#D45" target="_self" title="PCS_0:def.45">Defp5</a></i>;<br/>


</div>
