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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</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">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">ll</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">Sub</font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/>











<b>set </b><font color="Maroon">S'</font> =  <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font>;<br/>
<b>set </b><font color="Maroon">ll'</font> =  <a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">ll</font>,<font color="Maroon">Sub</font>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<b>reconsider </b><font color="Maroon">ll'</font> =  <a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">ll</font>,<font color="Maroon">Sub</font> as   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font> ;<br/>
<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/>

<a NAME="E3:22"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">J</font>,<font color="Maroon">v</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#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span> iff <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:22"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  iff <font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16">VALUAT_1:16</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E57</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16">VALUAT_1:16</a></i>;<br/>
<a NAME="E6:22"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</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">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font> iff <font color="Maroon">J</font>,<font color="Maroon">v</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">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:22"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</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#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span> iff <font color="Maroon">J</font>,<font color="Maroon">v</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">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">Sub</font> )
 <b>by </b><i>, , , , , <a class="ref" href="valuat_1.html#D12">VALUAT_1:def 12</a></i>;<br/>


</div>
