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<b>let </b><font color="Maroon" title="c3">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <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#K17" title="PCS_0:func.17">pcs-union</a> <font color="Olive" title="b1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">q</font> iff  ex <font color="Olive" title="b4">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b6">p'</font>, <font color="Olive" title="b7">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b5">P</font> st <br/>( <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b6">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b7">q'</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">C</font> be  <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <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#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font> iff  ex <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b5">p'</font>, <font color="Olive" title="b6">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">P</font> st <br/>( <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b6">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b5">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">q'</font> ) )</span><br/>



<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font>;<br/>
<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">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#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font> iff  ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">q'</font> ) )</span><br/>

<a NAME="E1:113"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:113"/>
( <font color="Maroon" title="c6">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font> implies  ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">q'</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">q'</font> ) implies <font color="Maroon" title="c6">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:113_1"/>
<font color="Maroon" title="c6">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">q'</font> )</span><br/>

<a NAME="E2:113_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</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="E3:113_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:113_1"/><i><font color="Green" title="E44">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <b>and </b><br/><a NAME="E6:113_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:113_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>and </b><br/><a NAME="E9:113_1"/><i><font color="Green" title="E47">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Z</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E6:113_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">I1</font> = <font color="Maroon" title="c3">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="pcs_0.html#E8:113_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A1</font> = <font color="Maroon" title="c4">C</font> as  <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c10">I1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">i1</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I1</font> <b>by </b><i><a class="txt" href="pcs_0.html#E8:113_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">P</font> = <font color="Maroon" title="c11">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c12">i1</font> as    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c9">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">q'</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c13">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b1">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">q'</font> )</span><br/>

<a NAME="E14:113_1"/>
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c11">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c12">i1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E9:113_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c15">q'</font> = <font color="Maroon" title="c7">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> <b>by </b><i><a class="txt" href="pcs_0.html#E5:113_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">p'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b1">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">q'</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c15">q'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">q'</font> )</span><br/>

<b>thus </b><a NAME="E16:113_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E8:113_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">q'</font> )</span><br/>

<b>thus </b><a NAME="E17:113_1"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">q'</font></span><br/>

<b>thus </b><a NAME="E18:113_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</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="c13">P</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E5:113_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="pcs_0.html#E9:113_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>given </b><font color="Maroon" title="c8">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> , <font color="Maroon" title="c10">p'</font>, <font color="Maroon" title="c11">q'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">P</font><b> such that </b><a NAME="E4:113"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>and </b><br/><a NAME="E5:113"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>and </b><br/><a NAME="E6:113"/><i><font color="Green" title="E46">A8</font></i>: 
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )
 <b>and </b><br/><a NAME="E7:113"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c10">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">q'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font></span><br/>

<a NAME="E8:113"/><i><font color="Green" title="E48">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p'</font>,<font color="Maroon" title="c11">q'</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="c9">P</font>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E7:113"><i><font color="Green" title="E47">A9</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">I1</font> = <font color="Maroon" title="c3">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="pcs_0.html#E4:113"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">I1</font> <b>by </b><i><a class="txt" href="pcs_0.html#E4:113"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">A1</font> = <font color="Maroon" title="c4">C</font> as  <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c12">I1</font> ;<br/>
<a NAME="E12:113"/>
<span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c14">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c14">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c13">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></i>;<br/>
<a NAME="E13:113"/><b>then </b>
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="c14">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:113"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E14:113"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E5:113"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="pcs_0.html#E6:113"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="pcs_0.html#E8:113"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:113"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
