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

<b>set </b><font color="Maroon" title="c5">P</font> =  <a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font>;<br/>
<b>set </b><font color="Maroon" title="c6">IR</font> = 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>;<br/>
<b>set </b><font color="Maroon" title="c7">TR</font> = the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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>;<br/>
<b>set </b><font color="Maroon" title="c8">CA</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>;<br/>
<a NAME="E1:120_1_1"/><i><font color="Green" title="E44">A1</font></i>: 
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> <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>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></i>;<br/>
<a NAME="E2:120_1_1"/><i><font color="Green" title="E45">A2</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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 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>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></i>;<br/>
<a NAME="E3:120_1_1"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">C</font> <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="E4:120_1_1"/>
 <a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font> is <a href="pcs_0.html#V13" title="PCS_0:attr.13">pcs-compatible</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">z</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="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a>,<a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</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 <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> or  not <font color="Maroon" title="c10">y</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 <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> or  not <font color="Maroon" title="c11">z</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 <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> or  not <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> or  not <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> or <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> )</span><br/>

<b>assume </b><a NAME="E1:120_1_1_1"/>
( <font color="Maroon" title="c9">x</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 <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> &amp; <font color="Maroon" title="c10">y</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 <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> &amp; <font color="Maroon" title="c11">z</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 <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"> (  not <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> or  not <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> or <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> )</span><br/>

<b>assume </b><a NAME="E2:120_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> or <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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> )</span><br/>

<b>then consider </b><font color="Maroon" title="c12">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:120_1_1_1"/><i><font color="Green" title="E47">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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="c12">Z1</font>
 <b>and </b><br/><a NAME="E5:120_1_1_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c12">Z1</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="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <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="E7:120_1_1_1"/><i><font color="Green" title="E49">A6</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#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="E8:120_1_1_1"/><i><font color="Green" title="E50">A7</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">Z1</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E5:120_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E9:120_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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></span><br/>

<b>then consider </b><font color="Maroon" title="c14">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:120_1_1_1"/><i><font color="Green" title="E51">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</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="c14">Z2</font>
 <b>and </b><br/><a NAME="E12:120_1_1_1"/><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c14">Z2</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="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:120_1_1_1"/><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c15">j</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="E15:120_1_1_1"/><i><font color="Green" title="E54">A11</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="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Z2</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E12:120_1_1_1"><i><font color="Green" title="E52">A9</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="c16">I</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#E7:120_1_1_1"><i><font color="Green" title="E49">A6</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="c17">C</font> = <font color="Maroon" title="c4">C</font> as   <a href="pcs_0.html#NM3" title="PCS_0:NM.3">pcs-Chain</a> of <font color="Maroon" title="c16">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">i</font> = <font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c19">j</font> = <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">I</font> <b>by </b><i><a class="txt" href="pcs_0.html#E7:120_1_1_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E19:120_1_1_1"/><i><font color="Green" title="E55">A12</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="c17">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</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="E20:120_1_1_1"/><b>then </b><i><font color="Green" title="E56">A13</font></i>: 
( <font color="Maroon" title="c9">x</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <font color="Maroon" title="c10">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E4:120_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="pcs_0.html#E8:120_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:120_1_1_1"/><i><font color="Green" title="E57">A14</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="c17">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</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="E22:120_1_1_1"/><i><font color="Green" title="E58">A15</font></i>: 
( <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</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> <font color="Maroon" title="c17">C</font> &amp; <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</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> <font color="Maroon" title="c17">C</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E23:120_1_1_1"/><i><font color="Green" title="E59">A16</font></i>: 
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></i>;<br/>
<a NAME="E24:120_1_1_1"/><i><font color="Green" title="E60">A17</font></i>: 
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_1_1_1"/>
( <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font> or <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font> )
 </span><b>by </b><i><a class="txt" href="pcs_0.html#E22:120_1_1_1"><i><font color="Green" title="E58">A15</font></i></a>, <a class="ref" href="pcs_0.html#D35" target="_self" title="PCS_0:def.35">Def35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:120_1_1_1_1_1"/>
<font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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></span><br/><div class="add"><a NAME="E2:120_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E61">A18</font></i>: 
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E3:120_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:120_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="pcs_0.html#E8:120_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1_1"><i><font color="Green" title="E55">A12</font></i></a></i>;<br/><a NAME="E4:120_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">x</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> &amp; <font color="Maroon" title="c10">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> &amp; <font color="Maroon" title="c11">z</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E11:120_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="pcs_0.html#E15:120_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="pcs_0.html#E21:120_1_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:120_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E62">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</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"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:120_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="pcs_0.html#E8:120_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="pcs_0.html#E11:120_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="pcs_0.html#E15:120_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1_1"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="pcs_0.html#E21:120_1_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="pcs_0.html#E24:120_1_1_1"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_1_1_1"><i><font color="Green" title="E61">A18</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><a NAME="E6:120_1_1_1_1_1"/>
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="pcs_0.html#E12:120_1_1_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="pcs_0.html#E15:120_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="pcs_0.html#E21:120_1_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E7:120_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</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="txt" href="pcs_0.html#E5:120_1_1_1_1_1"><i><font color="Green" title="E62">A19</font></i></a></i>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:120_1_1_1_1_2"/>
<font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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></span><br/><div class="add"><a NAME="E2:120_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E61">A20</font></i>: 
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E3:120_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</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"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E11:120_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="pcs_0.html#E15:120_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="pcs_0.html#E21:120_1_1_1"><i><font color="Green" title="E57">A14</font></i></a></i>;<br/><a NAME="E4:120_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">z</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 <span class="p1">(<span class="default"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:120_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E62">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</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"><font color="Maroon" title="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:120_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="pcs_0.html#E8:120_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="pcs_0.html#E11:120_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="pcs_0.html#E15:120_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1_1"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="pcs_0.html#E20:120_1_1_1"><i><font color="Green" title="E56">A13</font></i></a>, <a class="txt" href="pcs_0.html#E21:120_1_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="pcs_0.html#E23:120_1_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_1_1_2"><i><font color="Green" title="E61">A20</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><a NAME="E6:120_1_1_1_1_2"/>
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="c17">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="pcs_0.html#E5:120_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="pcs_0.html#E8:120_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1_1"><i><font color="Green" title="E55">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E7:120_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</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="txt" href="pcs_0.html#E5:120_1_1_1_1_2"><i><font color="Green" title="E62">A21</font></i></a></i>; <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></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">p'</font>, <font color="Maroon" title="c11">q</font>, <font color="Maroon" title="c12">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>; <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c10">p'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c12">q'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">q</font> implies <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:120_1_1"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c11">q</font>
 <b>and </b><br/><a NAME="E6:120_1_1"/><i><font color="Green" title="E48">A23</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="c9">p</font>
 <b>and </b><br/><a NAME="E7:120_1_1"/><i><font color="Green" title="E49">A24</font></i>: 
<font color="Maroon" title="c12">q'</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="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/>

<a NAME="E8:120_1_1"/>
<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="c9">p</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="txt" href="pcs_0.html#E6:120_1_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:120_1_1"/><i><font color="Green" title="E50">A25</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="c9">p</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="c13">Z1</font>
 <b>and </b><br/><a NAME="E11:120_1_1"/><i><font color="Green" title="E51">A26</font></i>: 
<font color="Maroon" title="c13">Z1</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="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <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="E13:120_1_1"/><i><font color="Green" title="E52">A27</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#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="E14:120_1_1"/><i><font color="Green" title="E53">A28</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="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Z1</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E11:120_1_1"><i><font color="Green" title="E51">A26</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="c15">I</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#E13:120_1_1"><i><font color="Green" title="E52">A27</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="c16">C</font> = <font color="Maroon" title="c4">C</font> as   <a href="pcs_0.html#NM3" title="PCS_0:NM.3">pcs-Chain</a> of <font color="Maroon" title="c15">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">I</font> <b>by </b><i><a class="txt" href="pcs_0.html#E13:120_1_1"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:120_1_1"/><i><font color="Green" title="E54">A29</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></i>;<br/>
<a NAME="E19:120_1_1"/><i><font color="Green" title="E55">A30</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</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/>
<b>then reconsider </b><font color="Maroon" title="c18">pi</font> = <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c19">p'i</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:120_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q'</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 <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="txt" href="pcs_0.html#E7:120_1_1"><i><font color="Green" title="E49">A24</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:120_1_1"/><i><font color="Green" title="E56">A31</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q'</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> <font color="Maroon" title="c20">Z2</font>
 <b>and </b><br/><a NAME="E24:120_1_1"/><i><font color="Green" title="E57">A32</font></i>: 
<font color="Maroon" title="c20">Z2</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="c16">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E26:120_1_1"/><i><font color="Green" title="E58">A33</font></i>: 
<font color="Maroon" title="c21">j</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="c16">C</font></span>)</span>
 <b>and </b><br/><a NAME="E27:120_1_1"/><i><font color="Green" title="E59">A34</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Z2</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E24:120_1_1"><i><font color="Green" title="E57">A32</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="c22">j</font> = <font color="Maroon" title="c21">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">I</font> <b>by </b><i><a class="txt" href="pcs_0.html#E26:120_1_1"><i><font color="Green" title="E58">A33</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E29:120_1_1"/><i><font color="Green" title="E60">A35</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></i>;<br/>
<a NAME="E30:120_1_1"/><i><font color="Green" title="E61">A36</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</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="E31:120_1_1"/><b>then </b><i><font color="Green" title="E62">A37</font></i>: 
( <font color="Maroon" title="c12">q'</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> &amp; <font color="Maroon" title="c11">q</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">qj</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E33:120_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E5:120_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">Z3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E35:120_1_1"/><i><font color="Green" title="E63">A38</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">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> <font color="Maroon" title="c24">Z3</font>
 <b>and </b><br/><a NAME="E36:120_1_1"/><i><font color="Green" title="E64">A39</font></i>: 
<font color="Maroon" title="c24">Z3</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="c16">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E38:120_1_1"/><i><font color="Green" title="E65">A40</font></i>: 
<font color="Maroon" title="c25">k</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#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c16">C</font></span>)</span>
 <b>and </b><br/><a NAME="E39:120_1_1"/><i><font color="Green" title="E66">A41</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Z3</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E36:120_1_1"><i><font color="Green" title="E64">A39</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="c26">k</font> = <font color="Maroon" title="c25">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">I</font> <b>by </b><i><a class="txt" href="pcs_0.html#E38:120_1_1"><i><font color="Green" title="E65">A40</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E41:120_1_1"/><i><font color="Green" title="E67">A42</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="c16">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">pk</font> = <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c28">qk</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E43:120_1_1"/><i><font color="Green" title="E68">A43</font></i>: 
( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</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> <font color="Maroon" title="c16">C</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</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> <font color="Maroon" title="c16">C</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</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> <font color="Maroon" title="c16">C</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E44:120_1_1"/><i><font color="Green" title="E69">A44</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#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c16">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="E45:120_1_1"/><b>then </b><i><font color="Green" title="E70">A45</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E2:120_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="pcs_0.html#E18:120_1_1"><i><font color="Green" title="E54">A29</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E46:120_1_1"/><i><font color="Green" title="E71">A46</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E2:120_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="pcs_0.html#E29:120_1_1"><i><font color="Green" title="E60">A35</font></i></a>, <a class="txt" href="pcs_0.html#E44:120_1_1"><i><font color="Green" title="E69">A44</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E47:120_1_1"/><i><font color="Green" title="E72">A47</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E2:120_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="pcs_0.html#E36:120_1_1"><i><font color="Green" title="E64">A39</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_1_2"/>
( ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> ) or ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> ) or ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> ) or ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> ) or ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> ) or ( <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> ) )
 </span><b>by </b><i><a class="txt" href="pcs_0.html#E43:120_1_1"><i><font color="Green" title="E68">A43</font></i></a>, <a class="ref" href="pcs_0.html#D35" target="_self" title="PCS_0:def.35">Def35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_1"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_1"/><i><font color="Green" title="E73">A48</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_1"/><i><font color="Green" title="E74">A49</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_1"/><i><font color="Green" title="E75">A50</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_1"><i><font color="Green" title="E74">A49</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_1"/>
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_1"><i><font color="Green" title="E73">A48</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A51</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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a></i>;<br/><a NAME="E6:120_1_1_2_1"/><b>then </b>
<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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1_2_1"><i><font color="Green" title="E75">A50</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">p'k</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:120_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q'</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_1"><i><font color="Green" title="E75">A50</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">q'k</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:120_1_1_2_1"/><i><font color="Green" title="E77">A52</font></i>: 
( <font color="Maroon" title="c29">p'k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">pk</font> &amp; <font color="Maroon" title="c30">q'k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">qk</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_1"><i><font color="Green" title="E75">A50</font></i></a>, <a class="txt" href="pcs_0.html#E5:120_1_1_2_1"><i><font color="Green" title="E76">A51</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E11:120_1_1_2_1"/>
<font color="Maroon" title="c27">pk</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c28">qk</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E12:120_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c29">p'k</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">q'k</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1_2_1"><i><font color="Green" title="E77">A52</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E13:120_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'k</font>,<font color="Maroon" title="c30">q'k</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E14:120_1_1_2_1"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E47:120_1_1"><i><font color="Green" title="E72">A47</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_2"/><i><font color="Green" title="E73">A53</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_2"/><i><font color="Green" title="E74">A54</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_2"/><i><font color="Green" title="E75">A55</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_2"><i><font color="Green" title="E74">A54</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_2"/><i><font color="Green" title="E76">A56</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_2"><i><font color="Green" title="E73">A53</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_2"/>
<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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_2"><i><font color="Green" title="E75">A55</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">p'k</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:120_1_1_2_2"/><i><font color="Green" title="E77">A57</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q'</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E4:120_1_1_2_2"><i><font color="Green" title="E76">A56</font></i></a></i>;<br/><a NAME="E8:120_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q'</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1_2_2"><i><font color="Green" title="E75">A55</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">q'k</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:120_1_1_2_2"/><i><font color="Green" title="E78">A58</font></i>: 
( <font color="Maroon" title="c29">p'k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">pk</font> &amp; <font color="Maroon" title="c30">q'k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">qk</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_2"><i><font color="Green" title="E75">A55</font></i></a>, <a class="txt" href="pcs_0.html#E7:120_1_1_2_2"><i><font color="Green" title="E77">A57</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E11:120_1_1_2_2"/>
<font color="Maroon" title="c27">pk</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c28">qk</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E12:120_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c29">p'k</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">q'k</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1_2_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E13:120_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'k</font>,<font color="Maroon" title="c30">q'k</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E14:120_1_1_2_2"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E47:120_1_1"><i><font color="Green" title="E72">A47</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_3"/><i><font color="Green" title="E73">A59</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_3"/><i><font color="Green" title="E74">A60</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_3"/><i><font color="Green" title="E75">A61</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_3"><i><font color="Green" title="E74">A60</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_3"/><i><font color="Green" title="E76">A62</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_3"><i><font color="Green" title="E74">A60</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_3"/>
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_3"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E6:120_1_1_2_3"/><b>then </b><i><font color="Green" title="E77">A63</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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a></i>;<br/><a NAME="E7:120_1_1_2_3"/><b>then </b><i><font color="Green" title="E78">A64</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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1_2_3"><i><font color="Green" title="E75">A61</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">p'j</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q'j</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">pj</font> = <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E7:120_1_1_2_3"><i><font color="Green" title="E78">A64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:120_1_1_2_3"/><i><font color="Green" title="E79">A65</font></i>: 
( <font color="Maroon" title="c29">p'j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c31">pj</font> &amp; <font color="Maroon" title="c30">q'j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">qj</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_3"><i><font color="Green" title="E75">A61</font></i></a>, <a class="txt" href="pcs_0.html#E6:120_1_1_2_3"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E12:120_1_1_2_3"/>
<font color="Maroon" title="c31">pj</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c23">qj</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="pcs_0.html#E4:120_1_1_2_3"><i><font color="Green" title="E76">A62</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E13:120_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c29">p'j</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">q'j</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E11:120_1_1_2_3"><i><font color="Green" title="E79">A65</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E14:120_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'j</font>,<font color="Maroon" title="c30">q'j</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E15:120_1_1_2_3"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E46:120_1_1"><i><font color="Green" title="E71">A46</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_4"/><i><font color="Green" title="E73">A66</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_4"/><i><font color="Green" title="E74">A67</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_4"/><i><font color="Green" title="E75">A68</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_4"><i><font color="Green" title="E74">A67</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_4"/><i><font color="Green" title="E76">A69</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_4"><i><font color="Green" title="E74">A67</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_4"/><i><font color="Green" title="E77">A70</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_4"><i><font color="Green" title="E73">A66</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E6:120_1_1_2_4"/><i><font color="Green" title="E78">A71</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="c9">p</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"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_4"><i><font color="Green" title="E75">A68</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">p'j</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q'j</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">pj</font> = <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E6:120_1_1_2_4"><i><font color="Green" title="E78">A71</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:120_1_1_2_4"/>
<font color="Maroon" title="c11">q</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">qi</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_4"><i><font color="Green" title="E73">A66</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><a NAME="E12:120_1_1_2_4"/><i><font color="Green" title="E79">A72</font></i>: 
( <font color="Maroon" title="c29">p'j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c31">pj</font> &amp; <font color="Maroon" title="c30">q'j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">qj</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_4"><i><font color="Green" title="E75">A68</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E13:120_1_1_2_4"/>
<font color="Maroon" title="c18">pi</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c32">qi</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="pcs_0.html#E5:120_1_1_2_4"><i><font color="Green" title="E77">A70</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E14:120_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c31">pj</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c23">qj</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:120_1_1_2_4"><i><font color="Green" title="E76">A69</font></i></a>, <a class="ref" href="pcs_0.html#T9" target="_self" title="PCS_0:th.9">Th9</a></i>;<br/><a NAME="E15:120_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c29">p'j</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">q'j</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E12:120_1_1_2_4"><i><font color="Green" title="E79">A72</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E16:120_1_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">p'j</font>,<font color="Maroon" title="c30">q'j</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E17:120_1_1_2_4"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E46:120_1_1"><i><font color="Green" title="E71">A46</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_5"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_5"/><i><font color="Green" title="E73">A73</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_5"/><i><font color="Green" title="E74">A74</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_5"/><i><font color="Green" title="E75">A75</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_5"><i><font color="Green" title="E74">A74</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_5"/><i><font color="Green" title="E76">A76</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_5"><i><font color="Green" title="E73">A73</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_5"/><i><font color="Green" title="E77">A77</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_5"><i><font color="Green" title="E74">A74</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">q'i</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E31:120_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_2_5"><i><font color="Green" title="E74">A74</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">qi</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E31:120_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_2_5"><i><font color="Green" title="E74">A74</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><a NAME="E8:120_1_1_2_5"/>
<font color="Maroon" title="c9">p</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 <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c31">pj</font> = <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_5"><i><font color="Green" title="E73">A73</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><a NAME="E10:120_1_1_2_5"/><i><font color="Green" title="E78">A78</font></i>: 
( <font color="Maroon" title="c19">p'i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">pi</font> &amp; <font color="Maroon" title="c29">q'i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c30">qi</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E5:120_1_1_2_5"><i><font color="Green" title="E77">A77</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E11:120_1_1_2_5"/>
<font color="Maroon" title="c31">pj</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c23">qj</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="pcs_0.html#E4:120_1_1_2_5"><i><font color="Green" title="E76">A76</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E12:120_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c18">pi</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">qi</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:120_1_1_2_5"><i><font color="Green" title="E75">A75</font></i></a>, <a class="ref" href="pcs_0.html#T9" target="_self" title="PCS_0:th.9">Th9</a></i>;<br/><a NAME="E13:120_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c19">p'i</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c29">q'i</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1_2_5"><i><font color="Green" title="E78">A78</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E14:120_1_1_2_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">p'i</font>,<font color="Maroon" title="c29">q'i</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E15:120_1_1_2_5"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E45:120_1_1"><i><font color="Green" title="E70">A45</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2_6"><b>suppose </b></a><b>that </b><a NAME="E1:120_1_1_2_6"/><i><font color="Green" title="E73">A79</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font>
 <b>and </b><br/><a NAME="E2:120_1_1_2_6"/><i><font color="Green" title="E74">A80</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font> <a href="pcs_0.html#R3" title="PCS_0:pred.3">c=</a> <font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q'</font></span><br/><div class="add"><a NAME="E3:120_1_1_2_6"/><i><font color="Green" title="E75">A81</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_6"><i><font color="Green" title="E74">A80</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E4:120_1_1_2_6"/><i><font color="Green" title="E76">A82</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:120_1_1_2_6"><i><font color="Green" title="E74">A80</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><a NAME="E5:120_1_1_2_6"/><i><font color="Green" title="E77">A83</font></i>: 
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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_6"><i><font color="Green" title="E73">A79</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">q'k</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E31:120_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="txt" href="pcs_0.html#E1:120_1_1_2_6"><i><font color="Green" title="E73">A79</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><a NAME="E7:120_1_1_2_6"/><i><font color="Green" title="E78">A84</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:120_1_1_2_6"><i><font color="Green" title="E73">A79</font></i></a>, <a class="ref" href="pcs_0.html#D34" target="_self" title="PCS_0:def.34">Def34</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">q'i</font> = <font color="Maroon" title="c12">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E31:120_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_2_6"><i><font color="Green" title="E74">A80</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">qi</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E31:120_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="txt" href="pcs_0.html#E2:120_1_1_2_6"><i><font color="Green" title="E74">A80</font></i></a>, <a class="txt" href="pcs_0.html#E7:120_1_1_2_6"><i><font color="Green" title="E78">A84</font></i></a>, <a class="txt" href="pcs_0.html#E47"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><a NAME="E10:120_1_1_2_6"/>
<font color="Maroon" title="c29">q'k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">qk</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E23:120_1_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="pcs_0.html#E27:120_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="pcs_0.html#E30:120_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="pcs_0.html#E5:120_1_1_2_6"><i><font color="Green" title="E77">A83</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E11:120_1_1_2_6"/><b>then </b><i><font color="Green" title="E79">A85</font></i>: 
( <font color="Maroon" title="c19">p'i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">pi</font> &amp; <font color="Maroon" title="c30">q'i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c31">qi</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:120_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pcs_0.html#E14:120_1_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="pcs_0.html#E19:120_1_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="txt" href="pcs_0.html#E4:120_1_1_2_6"><i><font color="Green" title="E76">A82</font></i></a>, <a class="ref" href="pcs_0.html#T8" target="_self" title="PCS_0:th.8">Th8</a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E12:120_1_1_2_6"/>
<font color="Maroon" title="c18">pi</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c31">qi</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E35:120_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="txt" href="pcs_0.html#E39:120_1_1"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="pcs_0.html#E41:120_1_1"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="pcs_0.html#E3:120_1_1_2_6"><i><font color="Green" title="E75">A81</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><a NAME="E13:120_1_1_2_6"/><b>then </b>
<font color="Maroon" title="c19">p'i</font> <a href="pcs_0.html#R2" title="PCS_0:pred.2">(--)</a> <font color="Maroon" title="c30">q'i</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E11:120_1_1_2_6"><i><font color="Green" title="E79">A85</font></i></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></i>;<br/><a NAME="E14:120_1_1_2_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">p'i</font>,<font color="Maroon" title="c30">q'i</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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">C</font> <a href="pcs_0.html#K16" title="PCS_0:func.16">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E15:120_1_1_2_6"/>
<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="c12">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</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="txt" href="pcs_0.html#E45:120_1_1"><i><font color="Green" title="E70">A45</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</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></div><b>end;</b></div>

</div>
