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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">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">s9</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b7">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<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="b5">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">v</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="Olive" title="b4">q</font>,<font color="Olive" title="b5">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">v</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">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">s9</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <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> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c3">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</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="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><span class="p1">(<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="b3">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b2">q</font>,<font color="Olive" title="b3">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font></span><br/><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">s9</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <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> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c3">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</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="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><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="Olive" title="b1">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b1">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s9</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <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> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c3">y</font> holds <br/> for <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b2">v</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="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><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="Olive" title="b2">v</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="Olive" title="b2">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">s9</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c3">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</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="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><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="Olive" title="b1">v</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="Olive" title="b1">v</font> )</span><br/>













<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ] <span class="kw">means </span> 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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> $1 <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>;<br/>
<div><span class="lab"><font color="Green" title="E29">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ] &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) )</span><br/><span class="kw">thus </span><a NAME="E1:40_1"/>
( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ] &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:40_1_1"/>
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c9">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c11">l</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c9">k</font><span class="kw"> such that </span><br/><a NAME="E2:40_1_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="qc_lang1.html#D16" title="QC_LANG1:def.16">QC_LANG1:def 16</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c13">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b2">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="Olive" title="b3">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c14">p</font>, <font color="Maroon" title="c15">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">y</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c13">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">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="Olive" title="b1">v</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="c15">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="Olive" title="b1">v</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:40_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1_1_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_1_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c13">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">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="Olive" title="b1">v</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="c15">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="Olive" title="b1">v</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">lx</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c17">ly</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>);<br/>
<span class="kw">let </span><font color="Maroon" title="c18">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font> implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">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="c18">v</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="c15">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="c18">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:40_1_1_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c18">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c14">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="c18">v</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="c15">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="c18">v</font></span><br/>

<a NAME="E7:40_1_1_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E2:40_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></span>;<br/>
<a NAME="E8:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T7" title="CQC_LANG:th.7">CQC_LANG:7</a></span>;<br/>
<a NAME="E9:40_1_1_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E2:40_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></span>;<br/>
<a NAME="E10:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T7" title="CQC_LANG:th.7">CQC_LANG:7</a></span>;<br/>
<a NAME="E11:40_1_1_1"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="cqc_lang.html#T7" title="CQC_LANG:th.7">CQC_LANG:7</a></span>;<br/>
<a NAME="E12:40_1_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="cqc_lang.html#T7" title="CQC_LANG:th.7">CQC_LANG:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">lx</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c12">x</font></span>)</span>), <font color="Maroon" title="c20">ly</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c11">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c13">y</font></span>)</span>) as   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c9">k</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E10:40_1_1_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E11:40_1_1_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></span>;<br/>
<a NAME="E14:40_1_1_1"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span><br/><a NAME="E3:40_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:40_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><div><span class="lab"><font color="Green" title="E46">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1_1_1"/>
<font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span><br/><a NAME="E2:40_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A19</font></span>: 
( <font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E4:40_1_1_1_1"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E3:40_1_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="cqc_lang.html#T3" title="CQC_LANG:th.3">CQC_LANG:3</a></span>;<br/><a NAME="E3:40_1_1_1_1_1"/>
<font color="Maroon" title="c18">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1_1_2"/>
<font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span><br/><a NAME="E2:40_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A20</font></span>: 
( <font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E4:40_1_1_1_1"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E3:40_1_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="cqc_lang.html#T3" title="CQC_LANG:th.3">CQC_LANG:3</a></span>;<br/><a NAME="E3:40_1_1_1_1_2"/>
<font color="Maroon" title="c18">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E2:40_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:40_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1_1_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:40_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#D3" target="_self" title="VALUAT_1:def.3">Def8</a></span>;<br/>
<a NAME="E17:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E15:40_1_1_1"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E43">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_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="c14">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="c18">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c15">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="c18">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><br/><a NAME="E2:40_1_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#K7" title="VALUAT_1:func.7">.</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="valuat_1.html#T8" target="_self" title="VALUAT_1:th.8">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_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="c15">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="c18">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E17:40_1_1_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="valuat_1.html#T8" target="_self" title="VALUAT_1:th.8">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">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="c18">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c15">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="c18">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="E2:40_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#K7" title="VALUAT_1:func.7">.</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="valuat_1.html#T7" target="_self" title="VALUAT_1:th.7">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">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="c18">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E17:40_1_1_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="valuat_1.html#T7" target="_self" title="VALUAT_1:th.7">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E20:40_1_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="c14">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="c18">v</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="c15">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="c18">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1_1"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b2">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="Olive" title="b3">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_2"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_2"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c13">v</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="c12">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="c13">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:40_1_2"/>
<font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c13">v</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="c12">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="c13">v</font></span><br/>

<a NAME="E5:40_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="cqc_lang.html#T15" title="CQC_LANG:th.15">CQC_LANG:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_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="c11">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="c13">v</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="c12">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="c13">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="cqc_lang.html#T15" title="CQC_LANG:th.15">CQC_LANG:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:40_1"/>
( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) &amp; ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_3"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 <span class="kw">and </span><br/><a NAME="E2:40_1_3"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:40_1_3"/>
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b2">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="Olive" title="b3">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_3_1"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_3_1"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font></span><br/>

<a NAME="E4:40_1_3_1"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="cqc_lang.html#T18" title="CQC_LANG:th.18">CQC_LANG:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">qa</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_3_1"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="cqc_lang.html#T8" title="CQC_LANG:th.8">CQC_LANG:8</a></span>;<br/>
<a NAME="E6:40_1_3_1"/><span class="lab"><font color="Green" title="E35">A30</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="cqc_lang.html#T18" title="CQC_LANG:th.18">CQC_LANG:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">pa</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="cqc_lang.html#T8" title="CQC_LANG:th.8">CQC_LANG:8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c15">v</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="c12">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="c15">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:40_1_3_1"/><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c15">v</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="c12">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="c15">v</font></span><br/>

<div><span class="lab"><font color="Green" title="E37">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_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="c11">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><br/><a NAME="E2:40_1_3_1_1"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pa</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="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E6:40_1_3_1"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="valuat_1.html#T10" target="_self" title="VALUAT_1:th.10">Th20</a></span>;<br/><a NAME="E3:40_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pa</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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T11" title="MARGREL1:th.11">MARGREL1:11</a></span>;<br/><a NAME="E4:40_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">qa</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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E8:40_1_3_1"><span class="lab"><font color="Green" title="E36">A31</font></span></a></span>;<br/><a NAME="E5:40_1_3_1_1"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">qa</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="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T11" title="MARGREL1:th.11">MARGREL1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:40_1_3_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="c12">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_3_1"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E4:40_1_3_1"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="valuat_1.html#T10" target="_self" title="VALUAT_1:th.10">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_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="c11">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="c15">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">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="c15">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="E2:40_1_3_1_2"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pa</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="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E6:40_1_3_1"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="valuat_1.html#T10" target="_self" title="VALUAT_1:th.10">Th20</a></span>;<br/><a NAME="E3:40_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pa</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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T11" title="MARGREL1:th.11">MARGREL1:11</a></span>;<br/><a NAME="E4:40_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">qa</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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E8:40_1_3_1"><span class="lab"><font color="Green" title="E36">A31</font></span></a></span>;<br/><a NAME="E5:40_1_3_1_2"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">qa</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="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T11" title="MARGREL1:th.11">MARGREL1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:40_1_3_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="c12">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_3_1"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E4:40_1_3_1"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="valuat_1.html#T10" target="_self" title="VALUAT_1:th.10">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:40_1_3_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c15">v</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="c12">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_3_1"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:40_1"/>
( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_4"/><span class="lab"><font color="Green" title="E30">A33</font></span>: 
<font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">and </span><br/><a NAME="E2:40_1_4"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_4"/><span class="lab"><font color="Green" title="E32">A35</font></span>: 
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:40_1_4"/>
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b2">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="Olive" title="b3">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_4_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_4_1"/><span class="lab"><font color="Green" title="E33">A36</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_4_1"/><span class="lab"><font color="Green" title="E34">A37</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="Olive" title="b1">v</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="c12">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="Olive" title="b1">v</font></span><br/>

<a NAME="E4:40_1_4_1"/><span class="lab"><font color="Green" title="E35">A38</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="ref" href="cqc_lang.html#T20" title="CQC_LANG:th.20">CQC_LANG:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">pla</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c14">pra</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="ref" href="cqc_lang.html#T9" title="CQC_LANG:th.9">CQC_LANG:9</a></span>;<br/>
<a NAME="E6:40_1_4_1"/><span class="lab"><font color="Green" title="E36">A39</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="ref" href="cqc_lang.html#T20" title="CQC_LANG:th.20">CQC_LANG:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">qla</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c16">qra</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="ref" href="cqc_lang.html#T9" title="CQC_LANG:th.9">CQC_LANG:9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c17">v</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="c12">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="c17">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:40_1_4_1"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
<font color="Maroon" title="c17">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c17">v</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="c12">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="c17">v</font></span><br/>

<div><span class="lab"><font color="Green" title="E38">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_1_1"/><span class="lab"><font color="Green" title="E39">A42</font></span>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c17">v</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="c12">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="c17">v</font></span><br/><div><span class="lab"><font color="Green" title="E40">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_1_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="c13">pla</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c17">v</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="c12">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="c17">v</font></span><br/><a NAME="E2:40_1_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">qla</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E8:40_1_4_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a></span>;<br/><a NAME="E3:40_1_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">qla</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="c17">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">qra</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="c17">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_4_1_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="c11">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="c17">v</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="c12">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="c17">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="txt" href="#E6:40_1_4_1"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="txt" href="#E1:40_1_4_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="valuat_1.html#T12" target="_self" title="VALUAT_1:th.12">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E41">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_1_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="c14">pra</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c11">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="c17">v</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="c12">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="c17">v</font></span><br/><a NAME="E2:40_1_4_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">qra</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E8:40_1_4_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a></span>;<br/><a NAME="E3:40_1_4_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">qla</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="c17">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">qra</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="c17">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_4_1_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="c11">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="c17">v</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="c12">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="c17">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="txt" href="#E6:40_1_4_1"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="txt" href="#E1:40_1_4_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="valuat_1.html#T12" target="_self" title="VALUAT_1:th.12">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:40_1_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">pla</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="c17">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pra</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="c17">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="txt" href="#E4:40_1_4_1"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="txt" href="#E1:40_1_4_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="valuat_1.html#T12" target="_self" title="VALUAT_1:th.12">Th22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_1_4_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="c11">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="c17">v</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="c12">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="c17">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4_1_1"><span class="lab"><font color="Green" title="E40">A43</font></span></a>, <a class="txt" href="#E3:40_1_4_1_1"><span class="lab"><font color="Green" title="E41">A44</font></span></a>, <a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_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="c11">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="c17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">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="c17">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="E2:40_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">pla</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="c17">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pra</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="c17">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="txt" href="#E4:40_1_4_1"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="ref" href="valuat_1.html#T12" target="_self" title="VALUAT_1:th.12">Th22</a></span>;<br/><a NAME="E3:40_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c14">pra</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>;<br/><a NAME="E4:40_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">qra</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E8:40_1_4_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a></span>;<br/><a NAME="E5:40_1_4_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="c13">pla</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4_1_2"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>;<br/><a NAME="E6:40_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">qla</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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E8:40_1_4_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a></span>;<br/><a NAME="E7:40_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c15">qla</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="c17">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">qra</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="c17">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_4_1_2"><span class="lab"><font color="Green" title="E40">A46</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:40_1_4_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="c12">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="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="txt" href="#E6:40_1_4_1"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="ref" href="valuat_1.html#T12" target="_self" title="VALUAT_1:th.12">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:40_1_4_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c11">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="c17">v</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="c12">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="c17">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E5:40_1"/>
( <font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_5"/><span class="lab"><font color="Green" title="E30">A47</font></span>: 
<font color="Maroon" title="c8">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 <span class="kw">and </span><br/><a NAME="E2:40_1_5"/><span class="lab"><font color="Green" title="E31">A48</font></span>: 
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">xx</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>, <font color="Maroon" title="c10">pp</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_5"/><span class="lab"><font color="Green" title="E32">A49</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c9">xx</font>,<font color="Maroon" title="c10">pp</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></span>;<br/>
<a NAME="E5:40_1_5"/><span class="lab"><font color="Green" title="E33">A50</font></span>: 
<font color="Maroon" title="c9">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="txt" href="#E4:40_1_5"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="ref" href="qc_lang1.html#D25" title="QC_LANG1:def.25">QC_LANG1:def 25</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:40_1_5"/>
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Olive" title="b2">y</font> holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</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="Olive" title="b4">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="Olive" title="b5">v</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font> holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> holds <br/><span class="p1">(<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="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</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="Olive" title="b2">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="Olive" title="b3">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="Olive" title="b1">v</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="c14">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="Olive" title="b1">v</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_5_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_5_1"/><span class="lab"><font color="Green" title="E34">A51</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_5_1"/><span class="lab"><font color="Green" title="E35">A52</font></span>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="Olive" title="b1">v</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="c14">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="Olive" title="b1">v</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="c15">v</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="c14">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="c15">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:40_1_5_1"/><span class="lab"><font color="Green" title="E36">A53</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c13">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="c15">v</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="c14">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="c15">v</font></span><br/>

<div><span class="lab"><font color="Green" title="E37">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_1"/><span class="lab"><font color="Green" title="E38">A55</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c13">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="c15">v</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="c14">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="c15">v</font></span><br/><a NAME="E2:40_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">ps</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_5_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="cqc_lang.html#T13" title="CQC_LANG:th.13">CQC_LANG:13</a></span>;<br/><a NAME="E4:40_1_5_1_1"/><span class="lab"><font color="Green" title="E39">A56</font></span>: 
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c16">ps</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="txt" href="#E2:40_1_5_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></span>;<br/><div><span class="lab"><font color="Green" title="E40">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_1_1"/><span class="lab"><font color="Green" title="E41">A58</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c13">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="c15">v</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="c14">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="c15">v</font></span><br/><a NAME="E2:40_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">qs</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_5_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="ref" href="cqc_lang.html#T13" title="CQC_LANG:th.13">CQC_LANG:13</a></span>;<br/><a NAME="E4:40_1_5_1_1_1"/><span class="lab"><font color="Green" title="E42">A59</font></span>: 
 <a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c17">qs</font>)</span>)</span>,<font color="Maroon" title="c6">J</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="valuat_1.html#E10"><span class="lab"><font color="Green" title="E8">Lm1</font></span></a></span>;<br/><a NAME="E5:40_1_5_1_1_1"/><span class="lab"><font color="Green" title="E43">A60</font></span>: 
 <a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c16">ps</font>)</span>)</span>,<font color="Maroon" title="c6">J</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">ps</font>,<font color="Maroon" title="c6">J</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="valuat_1.html#E10"><span class="lab"><font color="Green" title="E8">Lm1</font></span></a></span>;<br/><a NAME="E6:40_1_5_1_1_1"/><span class="lab"><font color="Green" title="E44">A61</font></span>: 
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c17">qs</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="txt" href="#E3:40_1_5_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></span>;<br/><div><span class="lab"><font color="Green" title="E45">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A63</font></span>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="c15">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c14">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="c15">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="E2:40_1_5_1_1_1_1"/>
 for <font color="Olive" title="b1">v1</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st (  for <font color="Olive" title="b2">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</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="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">v1</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> ) implies <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</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="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_5_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A64</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c17">qs</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="c18">v1</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="E2:40_1_5_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A65</font></span>: 
( <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5_1_1"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a></span>;<br/>
<a NAME="E3:40_1_5_1_1_1_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="c16">ps</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="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5_1_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E5:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E43">A60</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E46">A63</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="ref" href="valuat_1.html#T3" target="_self" title="VALUAT_1:th.3">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_1_5_1_1_1_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="c17">qs</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="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_5"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E4:40_1_5_1"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="txt" href="#E2:40_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_1_5_1_1_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="c14">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="txt" href="#E4:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="ref" href="valuat_1.html#T3" target="_self" title="VALUAT_1:th.3">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A66</font></span>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c14">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><br/><a NAME="E2:40_1_5_1_1_1_2"/>
 ex <font color="Olive" title="b1">v1</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; (  for <font color="Olive" title="b2">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</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="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c18">v1</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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><span class="kw"> such that </span><br/><a NAME="E2:40_1_5_1_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A67</font></span>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c16">ps</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="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">and </span><br/><a NAME="E3:40_1_5_1_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A68</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5_1_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E5:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E43">A60</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E46">A66</font></span></a>, <a class="ref" href="valuat_1.html#T2" target="_self" title="VALUAT_1:th.2">Th7</a></span>;<br/>
<a NAME="E4:40_1_5_1_1_1_2_1"/>
( <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5_1_1"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="txt" href="#E3:40_1_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A68</font></span></a></span>;<br/>
<a NAME="E5:40_1_5_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</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="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_5"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E4:40_1_5_1"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="txt" href="#E2:40_1_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E47">A67</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_1_5_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">v1</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; (  for <font color="Olive" title="b2">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</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="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A68</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_1_5_1_1_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="c14">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="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="txt" href="#E4:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="ref" href="valuat_1.html#T2" target="_self" title="VALUAT_1:th.2">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:40_1_5_1_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="c13">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="c15">v</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="c14">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_5_1_1_1"><span class="lab"><font color="Green" title="E45">A62</font></span></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_1_2"/><span class="lab"><font color="Green" title="E41">A69</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c13">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="c15">v</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="c14">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="c15">v</font></span><br/><a NAME="E2:40_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c10">pp</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E5:40_1_5"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E3:40_1_5_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="ref" href="cqc_lang.html#T24" title="CQC_LANG:th.24">CQC_LANG:24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_5_1_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="c13">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="c15">v</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="c14">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E5:40_1_5"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:40_1_5_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E1:40_1_5_1_1_2"><span class="lab"><font color="Green" title="E41">A69</font></span></a>, <a class="ref" href="cqc_lang.html#T27" title="CQC_LANG:th.27">CQC_LANG:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:40_1_5_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="c13">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="c15">v</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="c14">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_5_1_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_1_2"/><span class="lab"><font color="Green" title="E38">A70</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c13">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="c15">v</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="c14">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="c15">v</font></span><br/><a NAME="E2:40_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">pp</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E5:40_1_5"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:40_1_5_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="cqc_lang.html#T24" title="CQC_LANG:th.24">CQC_LANG:24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_5_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="c13">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="c15">v</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="c14">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_5"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E5:40_1_5"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E3:40_1_5_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="txt" href="#E1:40_1_5_1_2"><span class="lab"><font color="Green" title="E38">A70</font></span></a>, <a class="ref" href="cqc_lang.html#T27" title="CQC_LANG:th.27">CQC_LANG:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:40_1_5_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> (<font color="Maroon" title="c13">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="c15">v</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="c14">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="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_5_1"><span class="lab"><font color="Green" title="E37">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E2:40"/>
 for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S2" title="QC_LANG1:sch.2">QC_LANG1:sch 2</a>(<a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:40"/>
( <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> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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">s9</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c3">y</font> implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2: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>  st <font color="Olive" title="b1">v</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="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><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="Olive" title="b1">v</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="Olive" title="b1">v</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
