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<b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K23" title="PCS_0:func.23">--&gt;</a> <font color="Maroon" title="c4">Q</font></span>)</span><br/> for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font><br/> for <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Q</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">p1</font>,<font color="Olive" title="b5">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">p2</font>,<font color="Olive" title="b6">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b3">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">p2</font> holds <br/><font color="Olive" title="b5">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b6">q2</font></span><br/><b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K23" title="PCS_0:func.23">--&gt;</a> <font color="Maroon" title="c4">Q</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font><br/> for <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Q</font>  st <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b1">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">p2</font> holds <br/><font color="Olive" title="b3">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q2</font></span><br/><b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Q</font>  st <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Olive" title="b1">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Olive" title="b2">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c7">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">p2</font> holds <br/><font color="Olive" title="b1">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">q2</font></span><br/><b>let </b><font color="Maroon" title="c9">q1</font>, <font color="Maroon" title="c10">q2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">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="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c10">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c7">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">p2</font> implies <font color="Maroon" title="c9">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q2</font> )</span><br/>







<b>assume </b><a NAME="E1:186"/><i><font color="Green" title="E64">A1</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c10">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> or  not <font color="Maroon" title="c7">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">p2</font> or <font color="Maroon" title="c9">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q2</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c11">r1</font> = <font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c12">r2</font> = <font color="Maroon" title="c8">p2</font> as   <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> <b>by </b><i><a class="ref" href="pcs_0.html#D40" target="_self" title="PCS_0:def.40">Def40</a></i>;<br/>
<b>assume </b><a NAME="E3:186"/>
<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="c6">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"><font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K23" title="PCS_0:func.23">--&gt;</a> <font color="Maroon" title="c4">Q</font></span>)</span>
 ; <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"> (  not <font color="Maroon" title="c7">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">p2</font> or <font color="Maroon" title="c9">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c15">s</font>, <font color="Maroon" title="c16">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:186"/><i><font color="Green" title="E65">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E6:186"/>
( <font color="Maroon" title="c13">a</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> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c15">s</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> &amp; <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</font> )
 <b>and </b><br/><a NAME="E7:186"/><i><font color="Green" title="E66">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">s</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> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c16">t</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="c4">Q</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/>
<a NAME="E8:186"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q1</font> &amp; <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q2</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:186"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="pcs_0.html#E5:186"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:186"/><b>then </b>
( <font color="Maroon" title="c11">r1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c12">r2</font> or <font color="Maroon" title="c9">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q2</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E7:186"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E10:186"/>
(  not <font color="Maroon" title="c7">p1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">p2</font> or <font color="Maroon" title="c9">q1</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">q2</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#T34" target="_self" title="PCS_0:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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