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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">Sub</font> be   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>;<br/>

<b>set </b><font color="Maroon" title="c4">S1</font> = <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">S</font> = <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>;<br/>
<a NAME="E2:33"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>
 
<b>by </b><i><a class="ref" href="substut2.html#T22" target="_self" title="SUBSTUT2:th.22">Th22</a></i>;<br/>
<b>then </b><i><font color="Green" title="E22">A3</font></i>:  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span> = 
 <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>
<b>by </b><i><a class="ref" href="sublemma.html#D6" title="SUBLEMMA:def.6">SUBLEMMA:def 6</a></i>
<br/>.= 
<span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>
<b>by </b><i><a class="ref" href="substut2.html#T22" target="_self" title="SUBSTUT2:th.22">Th22</a></i>
;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
<span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>
 
<b>by </b><i><a class="txt" href="substut2.html#E2:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="substut1.html#D28" title="SUBSTUT1:def.28">SUBSTUT1:def 28</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
 <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sublemma.html#K12" title="SUBLEMMA:func.12">CQCQuant</a> <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>,<span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2">(<span class="default"><a href="sublemma.html#K11" title="SUBLEMMA:func.11">CQCSub_the_scope_of</a> <span class="p3"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T29" title="SUBLEMMA:th.29">SUBLEMMA:29</a></i>;<br/>
 <a href="sublemma.html#K11" title="SUBLEMMA:func.11">CQCSub_the_scope_of</a> <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> = 
 <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E3:33"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="substut2.html#E4:33"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="sublemma.html#D7" title="SUBLEMMA:def.7">SUBLEMMA:def 7</a></i>
<br/>.= 
<span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>,<font color="Maroon" title="c2">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> <a href="sublemma.html#K9" title="SUBLEMMA:func.9">`1</a> 
<b>by </b><i><a class="txt" href="substut2.html#E2:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="substut1.html#T21" title="SUBSTUT1:th.21">SUBSTUT1:21</a></i>
<br/>.= 
<span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
 <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sublemma.html#K12" title="SUBLEMMA:func.12">CQCQuant</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut2.html#E5:33"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="substut2.html#T22" target="_self" title="SUBSTUT2:th.22">Th22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">y</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">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p3">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span>;<br/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span> = 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5">(<span class="default"><a href="substut2.html#K7" title="SUBSTUT2:func.7">QScope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<span class="p5">(<span class="default"><a href="substut2.html#K8" title="SUBSTUT2:func.8">Qsc</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p4">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p0">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E2:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="substut2.html#E7:33"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="sublemma.html#T32" title="SUBLEMMA:th.32">SUBLEMMA:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p2">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p4">(<span class="default"><a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p0">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="substut2.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:33"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></i>;<br/>


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