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<div class="add">

<b>let </b><font color="Maroon" title="c3">P</font> be    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</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">D</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">P</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Olive" title="b1">D</font> iff ( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; ( for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b4">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b5">b</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c4">D</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font> iff ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b3">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">b</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font> iff ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</font> ) ) )</span><br/>





<b>thus </b><a NAME="E1:202"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font> implies ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</font> ) implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:202_1"/><i><font color="Green" title="E68">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</font> ) )</span><br/>

<b>hence </b><a NAME="E2:202_1"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#D46" target="_self" title="PCS_0:def.46">Def46</a></i>; <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</font></span><br/>

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c7">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<b>assume </b><a NAME="E3:202_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">b</font></span><br/>

<a NAME="E4:202_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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>
 
<b>by </b><i><a class="txt" href="#E1:202_1"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="pcs_0.html#D46" target="_self" title="PCS_0:def.46">Def46</a></i>;<br/>
<b>hence </b><a NAME="E5:202_1"/>
<font color="Maroon" title="c7">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</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>

<b>assume </b><b>that </b><br/><a NAME="E2:202"/><i><font color="Green" title="E68">A2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )
 <b>and </b><br/><a NAME="E3:202"/><i><font color="Green" title="E69">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">b</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E4:202"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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><br/>

<b>assume </b><a NAME="E1:202_2"/><i><font color="Green" title="E70">A4</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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><br/>

<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="#E2:202"><i><font color="Green" title="E68">A2</font></i></a></i>;<br/>
<a NAME="E3:202_2"/>
<font color="Maroon" title="c9">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c10">b</font>
 
<b>by </b><i><a class="txt" href="#E3:202"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="#E1:202_2"><i><font color="Green" title="E70">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:202_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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>
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</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>
<b>hence </b><a NAME="E5:202"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K25" title="PCS_0:func.25">pcs-general-power-TR</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="#E2:202"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="pcs_0.html#D46" target="_self" title="PCS_0:def.46">Def46</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>
