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

<b>set </b><font color="Maroon" title="c5">CA</font> =  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font>;<br/>
<b>set </b><font color="Maroon" title="c6">IRA</font> =  <a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <font color="Maroon" title="c4">C</font>;<br/>
<b>set </b><font color="Maroon" title="c7">TRA</font> =  <a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font>;<br/>
<b>set </b><font color="Maroon" title="c8">D</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">IR</font> =  <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>;<br/>
<b>set </b><font color="Maroon" title="c10">TR</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>;<br/>
<a NAME="E1:112_1_1"/><i><font color="Green" title="E41">b1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</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/>
<a NAME="E2:112_1_1"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:112_1_1_1"/>
<font color="Maroon" title="c11">x</font> <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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:112_1_1_1"/><i><font color="Green" title="E42">a1</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font>
 <b>and </b><br/><a NAME="E4:112_1_1_1"/><i><font color="Green" title="E43">a2</font></i>: 
<font color="Maroon" title="c12">P</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="c13">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:112_1_1_1"/><i><font color="Green" title="E44">a3</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="E7:112_1_1_1"/><i><font color="Green" title="E45">a4</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="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:112_1_1_1"><i><font color="Green" title="E43">a2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:112_1_1_1"/><i><font color="Green" title="E46">a8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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>then consider </b><font color="Maroon" title="c14">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E10:112_1_1_1"/><i><font color="Green" title="E47">a5</font></i>: 
( <font color="Maroon" title="c14">R</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="c13">i</font> &amp; <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="c13">i</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 <font color="Maroon" title="c14">R</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E6:112_1_1_1"><i><font color="Green" title="E44">a3</font></i></a>, <a class="ref" href="pcs_0.html#D5" target="_self" title="PCS_0:def.5">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:112_1_1_1"/><i><font color="Green" title="E48">a6</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:112_1_1_1"/><i><font color="Green" title="E49">a7</font></i>: 
( <font color="Maroon" title="c15">x1</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="c14">R</font> &amp; <font color="Maroon" title="c16">x2</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="c14">R</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:112_1_1_1"><i><font color="Green" title="E42">a1</font></i></a>, <a class="txt" href="pcs_0.html#E7:112_1_1_1"><i><font color="Green" title="E45">a4</font></i></a>, <a class="txt" href="pcs_0.html#E10:112_1_1_1"><i><font color="Green" title="E47">a5</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<a NAME="E14:112_1_1_1"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">S</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="c13">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</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="c13">i</font> <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 <font color="Olive" title="b1">S</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E6:112_1_1_1"><i><font color="Green" title="E44">a3</font></i></a>, <a class="txt" href="pcs_0.html#E8:112_1_1_1"><i><font color="Green" title="E46">a8</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E15:112_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">R</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:112_1_1"><i><font color="Green" title="E41">b1</font></i></a>, <a class="txt" href="pcs_0.html#E6:112_1_1_1"><i><font color="Green" title="E44">a3</font></i></a>, <a class="txt" href="pcs_0.html#E8:112_1_1_1"><i><font color="Green" title="E46">a8</font></i></a>, <a class="txt" href="pcs_0.html#E10:112_1_1_1"><i><font color="Green" title="E47">a5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:112_1_1_1"/><b>then </b>
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E13:112_1_1_1"><i><font color="Green" title="E49">a7</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="E17:112_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E12:112_1_1_1"><i><font color="Green" title="E48">a6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">IR</font> =  <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> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E4:112_1_1"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:112_1_1_2"/>
<font color="Maroon" title="c12">x</font> <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#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:112_1_1_2"/><i><font color="Green" title="E42">a1</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <b>and </b><br/><a NAME="E4:112_1_1_2"/><i><font color="Green" title="E43">a2</font></i>: 
<font color="Maroon" title="c13">P</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#K7" title="PCS_0:func.7">pcs-ToleranceRels</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="c14">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:112_1_1_2"/><i><font color="Green" title="E44">a3</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>and </b><br/><a NAME="E7:112_1_1_2"/><i><font color="Green" title="E45">a4</font></i>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</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="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:112_1_1_2"><i><font color="Green" title="E43">a2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:112_1_1_2"/><i><font color="Green" title="E46">a8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</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>then consider </b><font color="Maroon" title="c15">R</font> being    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> <b> such that </b><br/><a NAME="E10:112_1_1_2"/><i><font color="Green" title="E47">a5</font></i>: 
( <font color="Maroon" title="c15">R</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="c14">i</font> &amp; <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</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="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c15">R</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E6:112_1_1_2"><i><font color="Green" title="E44">a3</font></i></a>, <a class="ref" href="pcs_0.html#D19" target="_self" title="PCS_0:def.19">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:112_1_1_2"/><i><font color="Green" title="E48">a6</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:112_1_1_2"/><i><font color="Green" title="E49">a7</font></i>: 
( <font color="Maroon" title="c16">x1</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="c15">R</font> &amp; <font color="Maroon" title="c17">x2</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="c15">R</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:112_1_1_2"><i><font color="Green" title="E42">a1</font></i></a>, <a class="txt" href="pcs_0.html#E7:112_1_1_2"><i><font color="Green" title="E45">a4</font></i></a>, <a class="txt" href="pcs_0.html#E10:112_1_1_2"><i><font color="Green" title="E47">a5</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<a NAME="E14:112_1_1_2"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">S</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="c14">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</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="c14">i</font> <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 <font color="Olive" title="b1">S</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E6:112_1_1_2"><i><font color="Green" title="E44">a3</font></i></a>, <a class="txt" href="pcs_0.html#E8:112_1_1_2"><i><font color="Green" title="E46">a8</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E15:112_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">R</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:112_1_1"><i><font color="Green" title="E41">b1</font></i></a>, <a class="txt" href="pcs_0.html#E6:112_1_1_2"><i><font color="Green" title="E44">a3</font></i></a>, <a class="txt" href="pcs_0.html#E8:112_1_1_2"><i><font color="Green" title="E46">a8</font></i></a>, <a class="txt" href="pcs_0.html#E10:112_1_1_2"><i><font color="Green" title="E47">a5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:112_1_1_2"/><b>then </b>
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E13:112_1_1_2"><i><font color="Green" title="E49">a7</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="E17:112_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E12:112_1_1_2"><i><font color="Green" title="E48">a6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">TR</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<b>take </b>
 <a href="pcs_0.html#G2" title="PCS_0:aggr.2">pcs-Str</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">IR</font>,<font color="Maroon" title="c12">TR</font> #)
;<br/>

<b>thus </b><a NAME="E6:112_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pcs_0.html#G2" title="PCS_0:aggr.2">pcs-Str</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">IR</font>,<font color="Maroon" title="c12">TR</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="pcs_0.html#G2" title="PCS_0:aggr.2">pcs-Str</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">IR</font>,<font color="Maroon" title="c12">TR</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <a href="pcs_0.html#G2" title="PCS_0:aggr.2">pcs-Str</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">IR</font>,<font color="Maroon" title="c12">TR</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span> )
 ;<br/>


</div>
