<?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#K7" title="CQC_LANG:func.7">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>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">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#K15" title="CQC_LANG:func.15">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/>
<b>set </b><font color="Maroon" title="c5">B</font> = <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#K15" title="CQC_LANG:func.15">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>;<br/>
<a NAME="E1:32"/><i><font color="Green" title="E19">A1</font></i>: 
( <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#K15" title="CQC_LANG:func.15">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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K15" title="CQC_LANG:func.15">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> &amp; <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#K15" title="CQC_LANG:func.15">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="substut1.html#K23" title="SUBSTUT1:func.23">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<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="E2:32"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><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#K15" title="CQC_LANG:func.15">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> </span>)</span> <a href="substut2.html#K5" title="SUBSTUT2:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="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#K15" title="CQC_LANG:func.15">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> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><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#K15" title="CQC_LANG:func.15">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> </span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 
<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="E3:32"/>
<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#K15" title="CQC_LANG:func.15">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#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<a NAME="E4:32"/><b>then </b>
<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#K15" title="CQC_LANG:func.15">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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="substut2.html#K6" title="SUBSTUT2:func.6">CFQ</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:32"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><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#K15" title="CQC_LANG:func.15">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> </span>)</span> <a href="substut2.html#K5" title="SUBSTUT2:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K15" title="CQC_LANG:func.15">All</a> <span class="p3">(<span class="default"><span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><span class="p5"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p0">(<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="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#K15" title="CQC_LANG:func.15">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="substut1.html#K23" title="SUBSTUT1:func.23">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><span class="p0"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><font color="Maroon" title="c1">p</font>,<span class="p1">(<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="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#K15" title="CQC_LANG:func.15">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> </span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></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="txt" href="substut2.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="substut2.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<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#K15" title="CQC_LANG:func.15">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> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>
 
<b>by </b><i><a class="ref" href="substut1.html#D22" title="SUBSTUT1:def.22">SUBSTUT1:def 22</a></i>;<br/>
<a NAME="E7:32"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 
<b>by </b><i><a class="txt" href="substut2.html#E6:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="sublemma.html#D6" title="SUBLEMMA:def.6">SUBLEMMA:def 6</a></i>;<br/>
<a NAME="E8:32"/><b>then </b>
<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#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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><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#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="txt" href="substut2.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="substut2.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="substut2.html#E6:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="substut1.html#D24" title="SUBSTUT1:def.24">SUBSTUT1:def 24</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
( <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#K15" title="CQC_LANG:func.15">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#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> &amp;  <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="txt" href="substut2.html#E6:32"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>


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