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

<b>set </b><font color="Maroon" title="c5">R</font> = <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">TR</font> = the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>;<br/>
<a NAME="E1:78_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</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#D3" title="RELAT_2:def.3">RELAT_2:def 3</a>,<a class="ref" href="pcs_0.html#D11" target="_self" title="PCS_0:def.11">PCS_0:def 11</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="c7">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"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> or  not <font color="Maroon" title="c8">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"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> )</span><br/>

<b>assume </b><a NAME="E2:78_1_1"/>
( <font color="Maroon" title="c7">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"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> &amp; <font color="Maroon" title="c8">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"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> )</span><br/>

<b>assume </b><a NAME="E3:78_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></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="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span></span><br/>

<a NAME="E4:78_1_1"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:78_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:78_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="pcs_0.html#E3:78_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</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>
