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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b3">A</font><br/> for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b3">A</font> holds <br/> ( <font color="Olive" title="b4">J</font>,<font color="Olive" title="b5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> iff  ex <font color="Olive" title="b6">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">A</font> st <br/>( <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a</font> &amp; <font color="Olive" title="b4">J</font>,<font color="Olive" title="b5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b6">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b1">A</font> holds <br/> ( <font color="Olive" title="b2">J</font>,<font color="Olive" title="b3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> iff  ex <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> st <br/>( <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b2">J</font>,<font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b4">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c4">A</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c4">A</font> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> iff  ex <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> st <br/>( <font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c4">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c4">A</font> holds <br/> ( <font color="Maroon" title="c5">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> iff  ex <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Maroon" title="c5">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )</span><br/><b>let </b><font color="Maroon" title="c6">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c4">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> iff  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> st <br/>( <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )</span><br/>











<a NAME="E1:40"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <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"><font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> iff <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</font>,<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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA: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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T93" title="SUBLEMMA:th.93">SUBLEMMA:93</a></i>;<br/>
<a NAME="E2:40"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</font>,<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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA: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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> iff <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</font>,<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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="calcul_1.html#T23" target="_self" title="CALCUL_1:th.23">Th23</a></i>;<br/>
<a NAME="E3:40"/>
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">v</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="substut2.html#K5" title="SUBSTUT2:func.5">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D2" title="SUBLEMMA:def.2">SUBLEMMA:def 2</a></i>;<br/>
<a NAME="E4:40"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">v</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut2.html#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</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="E5:40"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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> <font color="Maroon" title="c6">v</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E6:40"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 
;<br/>
<a NAME="E7:40"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T59" title="SUBLEMMA:th.59">SUBLEMMA:59</a></i>;<br/>
<a NAME="E8:40"/><b>then </b>
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c6">v</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#K3" title="SUBSTUT2:func.3">Sbst</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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> <font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E5:40"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="funcop_1.html#T23" title="FUNCOP_1:th.23">FUNCOP_1:23</a></i>;<br/>
<b>hence </b><a NAME="E9:40"/>
( <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> iff  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> st <br/>( <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:40"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="calcul_1.html#E2:40"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="substut2.html#D2" title="SUBSTUT2:def.2">SUBSTUT2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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