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<b>let </b><font color="Maroon" title="c1">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">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><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="c1">A</font><br/> for <font color="Olive" title="b3">p</font> being   <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/> 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="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b4">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><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>; <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="c1">A</font><br/> for <font color="Olive" title="b2">p</font> being   <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/> for <font color="Olive" title="b3">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b3">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">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="c1">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">p</font> being   <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/> 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="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><br/><b>let </b><font color="Maroon" title="c4">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">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</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="c1">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="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><br/>









<div><i><font color="Green" title="E17">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/><a NAME="E2:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <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="c4">p</font></span>)</span>,<font color="Maroon" title="c5">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self" title="VALUAT_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E24"><i><font color="Green" title="E8">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <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="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font></span><br/><a NAME="E2:27_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <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="c4">p</font></span>)</span>,<font color="Maroon" title="c5">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E24"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self" title="VALUAT_1:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
( <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">p</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E17">A1</font></i></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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