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<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  <a href="pcs_0.html#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E1:168_1_1"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
 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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D40" target="_self" title="PCS_0:def.40">Def40</a></span>;<br/>
<a NAME="E2:168_1_1"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
 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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D40" target="_self" title="PCS_0:def.40">Def40</a></span>;<br/>
<a NAME="E3:168_1_1"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
 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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D40" target="_self" title="PCS_0:def.40">Def40</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">q9</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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c8">q9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">q</font> implies <font color="Maroon" title="c6">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">q9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:168_1_1"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:168_1_1"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p9</font>,<font color="Maroon" title="c5">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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:168_1_1"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a>,<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">q9</font></span><br/>

<a NAME="E7:168_1_1"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c6">p9</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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168_1_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">p90</font> = <font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c10">q90</font> = <font color="Maroon" title="c8">q9</font>, <font color="Maroon" title="c11">p0</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c12">q0</font> = <font color="Maroon" title="c7">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D40" target="_self" title="PCS_0:def.40">Def40</a></span>;<br/>
<a NAME="E9:168_1_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p0</font>,<font color="Maroon" title="c12">q0</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 <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:168_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E4:168_1_1"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:168_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A8</font></span>: 
 not <font color="Maroon" title="c11">p0</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">q0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></span>;<br/>
<a NAME="E11:168_1_1"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">p0</font>,<font color="Maroon" title="c9">p90</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:168_1_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E5:168_1_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/>
<a NAME="E12:168_1_1"/><span class="lab"><font color="Green" title="E72">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">q0</font>,<font color="Maroon" title="c10">q90</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:168_1_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E6:168_1_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/>
<a NAME="E13:168_1_1"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
<font color="Maroon" title="c11">p0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">p90</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:168_1_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E14:168_1_1"/>
<font color="Maroon" title="c12">q0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">q90</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168_1_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E15:168_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">p90</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q90</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:168_1_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E13:168_1_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="pcs_0.html#D22" target="_self" title="PCS_0:def.22">Def22</a></span>;<br/>
<a NAME="E16:168_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A12</font></span>: 
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">p90</font>,<font color="Maroon" title="c10">q90</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 <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></span>;<br/>
<a NAME="E17:168_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p9</font>,<font color="Maroon" title="c8">q9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:168_1_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E7:168_1_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:168_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p9</font>,<font color="Maroon" title="c8">q9</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#K21" title="PCS_0:func.21">pcs-reverse</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:168_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E16:168_1_1"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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