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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>









<b>assume </b><a NAME="E1:107"/><i><font color="Green">E90</font></i>: 
( <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="substut1.html#K39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K11" target="_self">CQCSub_the_scope_of</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:107"/><i><font color="Green">E91</font></i>: 
 <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font> is <a href="substut1.html#V7">Sub_universal</a>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sublemma.html#T28" target="_self">Th28</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E3:107"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="sublemma.html#K12" target="_self">CQCQuant</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2">(<span class="default"><a href="sublemma.html#K11" target="_self">CQCSub_the_scope_of</a> <span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="sublemma.html#K12" target="_self">CQCQuant</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sublemma.html#T31" target="_self">Th31</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="substut1.html#K35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:107"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p2">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E5:107"/><i><font color="Green">E94</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font> ) iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font> ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:107_1_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E2:107_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107_1_1"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:107_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:107_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:107_1_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E2:107_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107_1_2"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:107_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E6:107"/><i><font color="Green">E95</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sublemma.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E7:107"/><i><font color="Green">E96</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sublemma.html#T72" target="_self">Th72</a></i>;<br/>
<a NAME="E8:107"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_2"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:107_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:107_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> 
 
<b>by </b><i><a class="ref" href="sublemma.html#T76" target="_self">Th76</a></i>;<br/>
<b>hence </b><a NAME="E3:107_2_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T51" target="_self">Th51</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:107_2"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:107_2_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 ;<br/>

<a NAME="E2:107_2_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> 
 
<b>by </b><i><a class="ref" href="sublemma.html#T51" target="_self">Th51</a></i>;<br/>
<b>hence </b><a NAME="E3:107_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="sublemma.html#T76" target="_self">Th76</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:107"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:107"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sublemma.html#E2:107"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sublemma.html#E3:107"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sublemma.html#E4:107"><i><font color="Green">E93</font></i></a>, <a class="txt" href="sublemma.html#E5:107"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sublemma.html#E6:107"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sublemma.html#E7:107"><i><font color="Green">E96</font></i></a>, <a class="ref" href="sublemma.html#T29" target="_self">Th29</a>, <a class="ref" href="sublemma.html#T32" target="_self">Th32</a>, <a class="ref" href="sublemma.html#T57" target="_self">Th57</a>, <a class="ref" href="sublemma.html#T75" target="_self">Th75</a>, <a class="ref" href="sublemma.html#T91" target="_self">Th91</a></i>;<br/>


</div>
