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<span class="kw">let </span><font color="Maroon" title="c1">CX</font> be   <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">JH</font> being    <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Maroon" title="c1">CX</font><br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  holds <br/>( <font color="Olive" title="b1">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">JH</font> be    <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Maroon" title="c1">CX</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">p</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  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  holds <br/>( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</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"> (  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  implies ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c3">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c3">p</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a> 
 <span class="kw">and </span><br/><a NAME="E3:14"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</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">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c3">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c3">p</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E5:14"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E6:14"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> ex <font color="Olive" title="b2">r</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="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="substut2.html#T32" title="SUBSTUT2:th.32">SUBSTUT2:32</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>, <font color="Maroon" title="c6">r</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> <span class="kw"> such that </span><br/><a NAME="E8:14"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<a NAME="E9:14"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="substut2.html#T30" title="SUBSTUT2:th.30">SUBSTUT2:30</a></span>;<br/>
<a NAME="E10:14"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></span>;<br/>
<a NAME="E11:14"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E12:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">L</font> =  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>;<br/>
<a NAME="E13:14"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
1 <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>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="substut2.html#D5" title="SUBSTUT2:def.5">SUBSTUT2:def 5</a></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="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <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>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> implies  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) ) );<br/>
<a NAME="E14:14"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E15:14"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
 for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <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> ; <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">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:14_1"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:14_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<div><span class="lab"><font color="Green" title="E29">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">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 <br/>(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font> &amp;  ex <font color="Olive" title="b2">p1</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 <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p1</font> ) ) )</span><br/><a NAME="E2:14_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
 the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="substut2.html#D5" title="SUBSTUT2:def.5">SUBSTUT2:def 5</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c4">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c10">q</font> &amp;  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E3:14_1_1"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/><div><span class="lab"><font color="Green" title="E31">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1_1"/>
<font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)
 ; <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="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><br/><a NAME="E3:14_1_1_1"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goedelcp.html#T10" target="_self" title="GOEDELCP:th.10">Th10</a></span>;<br/><a NAME="E4:14_1_1_1"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a></span>;<br/><a NAME="E5:14_1_1_1"/><span class="kw">then </span>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goedelcp.html#T14" target="_self" title="GOEDELCP:th.14">Th14</a></span>;<br/><a NAME="E6:14_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:14_1_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="goedelcp.html#T8" target="_self" title="GOEDELCP:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:14_1_1_1"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="goedelcp.html#T3" target="_self" title="GOEDELCP:th.3">Th3</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="14_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1_2"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)
 ; <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">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><br/><a NAME="E3:14_1_1_2"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="goedelcp.html#T3" target="_self" title="GOEDELCP:th.3">Th3</a></span>;<br/><a NAME="E4:14_1_1_2"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a></span>;<br/><a NAME="E5:14_1_1_2"/><span class="kw">then </span>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goedelcp.html#T14" target="_self" title="GOEDELCP:th.14">Th14</a></span>;<br/><a NAME="E6:14_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:14_1_1_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="goedelcp.html#T8" target="_self" title="GOEDELCP:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:14_1_1_2"/>
<font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goedelcp.html#T10" target="_self" title="GOEDELCP:th.10">Th10</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="E6:14_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)</span>)</span> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c6">r</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:14_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="goedelcp.html#D1" target="_self" title="GOEDELCP:def.1">Def1</a>, <a class="ref" href="henmodel.html#D2" title="HENMODEL:def.2">HENMODEL:def 2</a>, <a class="ref" href="valuat_1.html#T17" title="VALUAT_1:th.17">VALUAT_1:17</a></span>;<br/><a NAME="E7:14_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c10">q</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c10">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="goedelcp.html#T11" target="_self" title="GOEDELCP:th.11">Th11</a>, <a class="ref" href="goedelcp.html#T12" target="_self" title="GOEDELCP:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:14_1_1"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E30">A17</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="14_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">  ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">p1</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 <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p1</font> ) )</span><br/><a NAME="E2:14_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E3:14_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:14_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">G1</font>, <font color="Maroon" title="c11">H1</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="E6:14_1_2"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
 the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:14_1_2"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
 the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">H1</font>
 <span class="kw">and </span><br/><a NAME="E8:14_1_2"/><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c10">G1</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c11">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:14_1_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="substut2.html#D5" title="SUBSTUT2:def.5">SUBSTUT2:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">p1</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> <span class="kw"> such that </span><br/><a NAME="E10:14_1_2"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">and </span><br/><a NAME="E11:14_1_2"/><span class="lab"><font color="Green" title="E35">A26</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font>
 <span class="kw">and </span><br/><a NAME="E12:14_1_2"/><span class="lab"><font color="Green" title="E36">A27</font></span>: 
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c12">p1</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c12">p1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E3:14_1_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:14_1_2"/><span class="lab"><font color="Green" title="E37">A28</font></span>: 
 ex <font color="Olive" title="b1">F3</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> st <br/>( <font color="Olive" title="b1">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">F3</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="substut2.html#T27" title="SUBSTUT2:th.27">SUBSTUT2:27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c11">H1</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="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="substut2.html#T28" title="SUBSTUT2:th.28">SUBSTUT2:28</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/><div><span class="lab"><font color="Green" title="E38">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_2_1"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c12">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/><a NAME="E2:14_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a></span>;<br/><a NAME="E3:14_1_2_1"/>
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c14">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E12:14_1_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="txt" href="#E1:14_1_2_1"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="goedelcp.html#D1" target="_self" title="GOEDELCP:def.1">Def1</a>, <a class="ref" href="henmodel.html#D2" title="HENMODEL:def.2">HENMODEL:def 2</a>, <a class="ref" href="valuat_1.html#T17" title="VALUAT_1:th.17">VALUAT_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:14_1_2_1"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E2:14_1_2_1"><span class="lab"><font color="Green" title="E40">A31</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="E16:14_1_2"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E13:14_1_2"><span class="lab"><font color="Green" title="E37">A28</font></span></a>, <a class="ref" href="substut2.html#T30" title="SUBSTUT2:th.30">SUBSTUT2:30</a></span>;<br/><div><span class="lab"><font color="Green" title="E40">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c15">F1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><span class="kw"> such that </span><a NAME="E2:14_1_2_2"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p1</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c15">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c16">F1</font> = <font color="Maroon" title="c15">F1</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="#E2:14_1_2_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="cqc_lang.html#T9" title="CQC_LANG:th.9">CQC_LANG:9</a></span>;<br/><a NAME="E4:14_1_2_2"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c16">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_2_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a></span>;<br/><a NAME="E5:14_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:14_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A36</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:14_1_2_2"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E5:14_1_2_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:14_1_2_2"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E6:14_1_2_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:14_1_2_2"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E7:14_1_2_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E10:14_1_2_2"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c16">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_2_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E8:14_1_2_2"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a></span>;<br/><a NAME="E11:14_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A39</font></span>: 
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c16">F1</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c16">F1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="goedelcp.html#T8" target="_self" title="GOEDELCP:th.8">Th8</a></span>;<br/><a NAME="E12:14_1_2_2"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E5:14_1_2_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:14_1_2_2"/>
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c14">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="txt" href="#E2:14_1_2_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E11:14_1_2_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="goedelcp.html#T6" target="_self" title="GOEDELCP:th.6">Th6</a>, <a class="ref" href="valuat_1.html#T18" title="VALUAT_1:th.18">VALUAT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E14:14_1_2_2"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E12:14_1_2_2"><span class="lab"><font color="Green" title="E47">A40</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="lab"><font color="Green" title="E41">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c15">F1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><span class="kw"> such that </span><a NAME="E2:14_1_2_3"/><span class="lab"><font color="Green" title="E42">A42</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F1</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c12">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c16">F1</font> = <font color="Maroon" title="c15">F1</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="#E2:14_1_2_3"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="ref" href="cqc_lang.html#T9" title="CQC_LANG:th.9">CQC_LANG:9</a></span>;<br/><a NAME="E4:14_1_2_3"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c16">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_2_3"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a></span>;<br/><a NAME="E5:14_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:14_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A45</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:14_1_2_3"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E5:14_1_2_3"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:14_1_2_3"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E6:14_1_2_3"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:14_1_2_3"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E7:14_1_2_3"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E10:14_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A48</font></span>: 
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c16">F1</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c16">F1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:14_1_2_3"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="txt" href="#E8:14_1_2_3"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="ref" href="goedelcp.html#T8" target="_self" title="GOEDELCP:th.8">Th8</a></span>;<br/><a NAME="E11:14_1_2_3"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E5:14_1_2_3"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:14_1_2_3"/>
( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> iff <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c14">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="txt" href="#E2:14_1_2_3"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E10:14_1_2_3"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="goedelcp.html#T6" target="_self" title="GOEDELCP:th.6">Th6</a>, <a class="ref" href="valuat_1.html#T18" title="VALUAT_1:th.18">VALUAT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E13:14_1_2_3"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E11:14_1_2_3"><span class="lab"><font color="Green" title="E49">A49</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="14_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c15">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><a NAME="E2:14_1_2_4"/><span class="lab"><font color="Green" title="E42">A50</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c12">p1</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:14_1_2_4"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:14_1_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:14_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E2:14_1_2_4"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</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="E20:14_1_2"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14_1_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E8:14_1_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E10:14_1_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="txt" href="#E15:14_1_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="txt" href="#E17:14_1_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E18:14_1_2"><span class="lab"><font color="Green" title="E41">A41</font></span></a>, <a class="ref" href="qc_lang2.html#D19" title="QC_LANG2:def.19">QC_LANG2:def 19</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="E6:14_1"/>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E29">A16</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="E16:14"/>
 for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E25">A12</font></span></a>);<br/></span>
<a NAME="E17:14"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p1</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 <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font></span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> ) &amp; ( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:14"/>
( <font color="Maroon" title="c2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c3">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c3">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="substut2.html#D5" title="SUBSTUT2:def.5">SUBSTUT2:def 5</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>
