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

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</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">A</font> being 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> <br/> for <font color="Olive" title="b2">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b4">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b5">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> holds <br/>( ( for <font color="Olive" title="b6">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span></span>)</span>,<font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b5">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b4">S</font> ) iff for <font color="Olive" title="b6">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b5">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b4">S</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  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> ; <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">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b4">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> holds <br/>( ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b4">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b5">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b4">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b5">a</font></span>)</span></span>)</span>,<font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b4">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b5">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S</font> ) iff for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b4">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b5">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b4">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b5">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S</font> )</span><br/><b>let </b><font color="Maroon" title="c3">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</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">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b3">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> holds <br/>( ( for <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b3">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b3">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span></span>)</span>,<font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b3">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S</font> ) iff for <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b3">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b3">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</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">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b2">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> holds <br/>( ( for <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span></span>)</span>,<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S</font> ) iff for <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S</font> )</span><br/><b>let </b><font color="Maroon" title="c5">S</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</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">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> holds <br/>( ( for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) iff for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> )</span><br/><b>let </b><font color="Maroon" title="c6">xSQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA: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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> implies ( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) iff for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) )</span><br/>











<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E56">A1</font></i>: 
<span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) iff for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> )</span><br/>

<b>set </b><font color="Maroon" title="c7">S1</font> =  <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font>;<br/>
<b>set </b><font color="Maroon" title="c8">z</font> =  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span>;<br/>
<b>thus </b><a NAME="E2:75"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</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> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1"/><i><font color="Green" title="E57">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</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> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font></span><br/>

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font></span><br/>

<a NAME="E2:75_1"/>
 <a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font>
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="sublemma.html#T56" target="_self" title="SUBLEMMA:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E3:75_1"/>
<font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="#E1:75_1"><i><font color="Green" title="E57">A2</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>

<b>thus </b><a NAME="E3:75"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_2"/><i><font color="Green" title="E57">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</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> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font></span><br/>

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font></span><br/>

<a NAME="E2:75_2"/>
 <a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font>
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="sublemma.html#T56" target="_self" title="SUBLEMMA:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E3:75_2"/>
<font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c6">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="#E1:75_2"><i><font color="Green" title="E57">A3</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>
