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

<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>, <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">p</font>, <font color="Olive" title="b4">q</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="b5">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b6">s</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b6">s</font> &amp; <font color="Olive" title="b5">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b5">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Olive" title="b4">q</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">p</font>, <font color="Olive" title="b2">q</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><br/> for <font color="Olive" title="b4">s</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b4">s</font> &amp; <font color="Olive" title="b3">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b3">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Olive" title="b2">q</font></span><br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</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><br/> for <font color="Olive" title="b2">s</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Olive" title="b1">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font></span><br/><b>let </b><font color="Maroon" title="c6">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"> for <font color="Olive" title="b1">s</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font></span><br/><b>let </b><font color="Maroon" title="c7">s</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> )
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green" title="E36">A2</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c7">s</font>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <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"> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font></span><br/><div><i><font color="Green" title="E39">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">u</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"> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</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>consider </b><font color="Maroon" title="c9">w</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><b> such that </b><br/><a NAME="E2:56_1_1"/><i><font color="Green" title="E40">A6</font></i>: 
( ( for <font color="Olive" title="b1">z</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon" title="c9">w</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> ) &amp; <font color="Maroon" title="c9">w</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> )
 <b>from </b><i><a class="ref" href="valuat_1.html#S1" target="_self" title="VALUAT_1:sch.1">VALUAT_1:sch 1</a>();<br/></i><a NAME="E3:56_1_1"/>
<font color="Maroon" title="c9">w</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:56_1_1"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/><a NAME="E4:56_1_1"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
<span class="p1">(<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="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:56"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="valuat_1.html#T42" target="_self" title="VALUAT_1:th.42">Th42</a></i>;<br/><a NAME="E5:56_1_1"/>
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c9">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:56"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="valuat_1.html#D13" target="_self" title="VALUAT_1:def.13">Def13</a></i>;<br/><a NAME="E6:56_1_1"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
<span class="p1">(<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="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</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/><a NAME="E7:56_1_1"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:56"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="valuat_1.html#E2:56"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="valuat_1.html#E1:56_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="valuat_1.html#T43" target="_self" title="VALUAT_1:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E8:56_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</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#E2:56_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="valuat_1.html#E4:56_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="valuat_1.html#E6:56_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="valuat_1.html#T39" target="_self" title="VALUAT_1:th.39">Th39</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="56_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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"> <font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c5">q</font></span><br/><a NAME="E1:56_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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#E2:56_1"><i><font color="Green" title="E39">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E2:56_1_2"/>
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c5">q</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="E4:56_1"/>
<font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D13" target="_self" title="VALUAT_1:def.13">Def13</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="E5:56"/>
<font color="Maroon" title="c6">J</font> <a href="valuat_1.html#R2" title="VALUAT_1:pred.2">|=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:56"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="valuat_1.html#E3:56"><i><font color="Green" title="E37">A3</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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