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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</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">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="E1:40"/><i><font color="Green">E52</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"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> iff <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p2"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span></span>)</span> <a href="sublemma.html#R1">|=</a> <span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T93">SUBLEMMA:93</a></i>;<br/>
<a NAME="E2:40"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p2"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span></span>)</span> <a href="sublemma.html#R1">|=</a> <span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> iff <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p2"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/>
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><span class="p3"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p4">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="substut2.html#K5">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#D2">SUBLEMMA:def 2</a></i>;<br/>
<a NAME="E4:40"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:40"/>
 <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E6:40"/><b>then </b>
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="relat_1.html#K5">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="substut2.html#D1">SUBSTUT2:def 1</a></i>;<br/>
<a NAME="E7:40"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="relat_1.html#K5">*</a> <font color="Maroon">v</font>
 
;<br/>
<a NAME="E8:40"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
;<br/>
<a NAME="E9:40"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T59">SUBLEMMA:59</a></i>;<br/>
<a NAME="E10:40"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T23">FUNCOP_1:23</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>;<br/>
<a NAME="E11:40"/>
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:40"/><b>then </b>
 <a href="sublemma.html#K3">Val_S</a> <font color="Maroon">v</font>,<span class="p1"><a href="substut2.html#K2">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3">Sbst</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="substut2.html#K2">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D9">SUBLEMMA:def 9</a></i>;<br/>
<b>hence </b><a NAME="E13:40"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> iff  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) )
 <b>by </b><i>, , <a class="ref" href="substut2.html#D2">SUBSTUT2:def 2</a></i>;<br/>


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