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<b>let </b><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#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">n</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 ( for <font color="Olive" title="b3">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &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="b3">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b3">p</font> ) ) holds <br/>for <font color="Olive" title="b3">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="b3">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b3">p</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</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 ( for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &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="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> ) ) holds <br/>for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="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/><b>let </b><font color="Maroon" title="c3">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &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> ) ) implies for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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/>

<b>let </b><font color="Maroon" title="c4">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> ) )</span><br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E16">A2</font></i>: 
(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> )</span><br/>

<a NAME="E3:15"/><i><font color="Green" title="E17">A3</font></i>: 
(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</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="Maroon" title="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> )</span><br/><a NAME="E2:15_1"/><b>then </b>
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">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c5">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="substut2.html#T34" title="SUBSTUT2:th.34">SUBSTUT2:34</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">L</font> being    <a href="substut2.html#M1" title="SUBSTUT2:mode.1">PATH</a> of <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">p</font>;<br/><a NAME="E6:15_1"/><i><font color="Green" title="E20">A6</font></i>: 
( 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> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c6">L</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> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; ( 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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c6">L</font> holds <br/> ex <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">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>  st <br/>( <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b3">H1</font> &amp; <font color="Olive" title="b2">G1</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Olive" title="b3">H1</font> ) ) )
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="substut2.html#D6" title="SUBSTUT2:def.6">SUBSTUT2:def 6</a></i>;<br/><b>defpred </b><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> ] <b>means </b>( 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> <font color="Maroon" title="c6">L</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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</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="c5">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="E7:15_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E8:15_1"/><i><font color="Green" title="E22">A8</font></i>: 
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="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>assume </b><a NAME="E2:15_1_1"/><i><font color="Green" title="E24">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c6">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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/>

<b>set </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<div><i><font color="Green" title="E25">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q</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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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="E2:15_1_1_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">q</font>
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="substut2.html#D6" title="SUBSTUT2:def.6">SUBSTUT2:def 6</a></i>;<br/><b>take </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c5">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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="E3:15_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> <font color="Maroon" title="c9">q</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c9">q</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="goedelcp.html#T15" target="_self" title="GOEDELCP:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:15_1_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E2:15_1_1_1"><i><font color="Green" title="E26">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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="E2:15_1_1_2"/><b>then </b>
(  <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="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b><i><font color="Green" title="E26">A13</font></i>: 
( <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="c7">k</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c6">L</font> )
 <b>by </b><i><a class="txt" href="#E2:15_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">G1</font>, <font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E5:15_1_1_2"/><i><font color="Green" title="E27">A14</font></i>: 
( <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> &amp; <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">H1</font> &amp; <font color="Maroon" title="c9">G1</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c10">H1</font> )
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="substut2.html#D6" title="SUBSTUT2:def.6">SUBSTUT2:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">p1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E7:15_1_1_2"/><i><font color="Green" title="E28">A15</font></i>: 
(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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="c11">p1</font> &amp; <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c11">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="Maroon" title="c11">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="Maroon" title="c11">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c11">p1</font> ) )
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E3:15_1_1_2"><i><font color="Green" title="E26">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">F3</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><b> such that </b><br/><a NAME="E9:15_1_1_2"/><i><font color="Green" title="E29">A16</font></i>: 
( <font color="Maroon" title="c12">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c12">F3</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E2:15_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="substut2.html#T27" title="SUBSTUT2:th.27">SUBSTUT2:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c10">H1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E2:15_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="substut2.html#T28" title="SUBSTUT2:th.28">SUBSTUT2:28</a></i>;<br/><b>take </b><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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><i><font color="Green" title="E30">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_1"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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="E2:15_1_1_2_1"/><b>then </b>
(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</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="Maroon" title="c14">s</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="Maroon" title="c14">s</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> ) )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="goedelcp.html#D1" target="_self" title="GOEDELCP:def.1">Def1</a>, <a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a>, <a class="ref" href="henmodel.html#D3" title="HENMODEL:def.3">HENMODEL:def 3</a>, <a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></i>;<br/><b>hence </b><a NAME="E3:15_1_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:15_1_1_2"/><i><font color="Green" title="E31">A18</font></i>: 
 <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> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E9:15_1_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="substut2.html#T30" title="SUBSTUT2:th.30">SUBSTUT2:30</a></i>;<br/><div><i><font color="Green" title="E32">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c15">F1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><b> such that </b><a NAME="E2:15_1_1_2_2"/><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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/><b>reconsider </b><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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="#E2:15_1_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></i>;<br/><a NAME="E4:15_1_1_2_2"/><i><font color="Green" title="E34">A21</font></i>: 
 <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="c11">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>
 <b>by </b><i><a class="txt" href="#E2:15_1_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a></i>;<br/><a NAME="E5:15_1_1_2_2"/><b>then </b>
1 <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> <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>
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:15_1_1_2_2"/><b>then </b>
1 <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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E12:15_1_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:15_1_1_2_2"/><b>then </b>
<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:15_1_1_2_2"/><b>then </b>
( <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> )
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/><a NAME="E9:15_1_1_2_2"/><b>then </b><i><font color="Green" title="E35">A22</font></i>: 
( <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</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="Maroon" title="c14">s</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="Maroon" title="c14">s</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> ) )
 <b>by </b><i><a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="#E2:15_1_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="goedelcp.html#T6" target="_self" title="GOEDELCP:th.6">Th6</a>, <a class="ref" href="valuat_1.html#T29" title="VALUAT_1:th.29">VALUAT_1:29</a></i>;<br/><a NAME="E10:15_1_1_2_2"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="#E4:15_1_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E11:15_1_1_2_2"/><b>then </b>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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>
 <b>by </b><i><a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:15_1_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E9:15_1_1_2_2"><i><font color="Green" title="E35">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E33">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c15">F1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><b> such that </b><a NAME="E2:15_1_1_2_3"/><i><font color="Green" title="E34">A24</font></i>: 
<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="c11">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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/><b>reconsider </b><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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="#E2:15_1_1_2_3"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></i>;<br/><a NAME="E4:15_1_1_2_3"/><i><font color="Green" title="E35">A25</font></i>: 
 <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="c11">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>
 <b>by </b><i><a class="txt" href="#E2:15_1_1_2_3"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a></i>;<br/><a NAME="E5:15_1_1_2_3"/><b>then </b>
1 <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> <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>
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:15_1_1_2_3"/><b>then </b>
1 <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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E12:15_1_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:15_1_1_2_3"/><b>then </b>
<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:15_1_1_2_3"/><b>then </b>
( <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> )
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/><a NAME="E9:15_1_1_2_3"/><b>then </b><i><font color="Green" title="E36">A26</font></i>: 
( <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</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="Maroon" title="c14">s</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="Maroon" title="c14">s</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> ) )
 <b>by </b><i><a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="#E2:15_1_1_2_3"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="goedelcp.html#T6" target="_self" title="GOEDELCP:th.6">Th6</a>, <a class="ref" href="valuat_1.html#T29" title="VALUAT_1:th.29">VALUAT_1:29</a></i>;<br/><a NAME="E10:15_1_1_2_3"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c11">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>
 <b>by </b><i><a class="txt" href="#E4:15_1_1_2_3"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E11:15_1_1_2_3"/><b>then </b>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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>
 <b>by </b><i><a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:15_1_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E9:15_1_1_2_3"><i><font color="Green" title="E36">A26</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c15">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><a NAME="E2:15_1_1_2_4"/><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c11">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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="E3:15_1_1_2_4"/><i><font color="Green" title="E35">A28</font></i>: 
 <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="c11">p1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:15_1_1_2_4"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></i>;<br/><a NAME="E4:15_1_1_2_4"/><b>then </b>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c11">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E12:15_1_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:15_1_1_2_4"/><b>then </b>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c11">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:15_1_1_2_4"/><b>then </b><i><font color="Green" title="E36">A29</font></i>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a></i>;<br/><a NAME="E7:15_1_1_2_4"/><i><font color="Green" title="E37">A30</font></i>: 
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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>
 <b>by </b><i><a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="#E3:15_1_1_2_4"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><i><font color="Green" title="E38">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_4_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#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c16">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:15_1_1_2_4_1"/><i><font color="Green" title="E39">A32</font></i>: 
<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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#T10" target="_self" title="GOEDELCP:th.10">Th10</a></i>;<br/><a NAME="E4:15_1_1_2_4_1"/>
 <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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E6:15_1_1_2_4"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="goedelcp.html#T14" target="_self" title="GOEDELCP:th.14">Th14</a></i>;<br/><a NAME="E5:15_1_1_2_4_1"/><b>then </b>
<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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:15_1_1_2_4_1"><i><font color="Green" title="E39">A32</font></i></a></i>;<br/><b>hence </b><a NAME="E6:15_1_1_2_4_1"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="goedelcp.html#T3" target="_self" title="GOEDELCP:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_4_2"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c16">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:15_1_1_2_4_2"/><i><font color="Green" title="E39">A33</font></i>: 
<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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="goedelcp.html#T3" target="_self" title="GOEDELCP:th.3">Th3</a></i>;<br/><a NAME="E4:15_1_1_2_4_2"/>
 <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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E6:15_1_1_2_4"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="goedelcp.html#T14" target="_self" title="GOEDELCP:th.14">Th14</a></i>;<br/><a NAME="E5:15_1_1_2_4_2"/><b>then </b>
<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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:15_1_1_2_4_2"><i><font color="Green" title="E39">A33</font></i></a></i>;<br/><b>hence </b><a NAME="E6:15_1_1_2_4_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#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#T10" target="_self" title="GOEDELCP:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:15_1_1_2_4"/><b>then </b>
( <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#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</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#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K13" title="CQC_LANG:func.13">Ex</a> <font color="Maroon" title="c15">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c11">p1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E8:15_1_1_2_4"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="goedelcp.html#D1" target="_self" title="GOEDELCP:def.1">Def1</a>, <a class="ref" href="henmodel.html#D3" title="HENMODEL:def.3">HENMODEL:def 3</a>, <a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></i>;<br/><a NAME="E11:15_1_1_2_4"/><b>then </b>
( <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &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="Maroon" title="c14">s</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</font> ) &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c14">s</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="Maroon" title="c14">s</font> ) )
 <b>by </b><i><a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E2:15_1_1_2_4"><i><font color="Green" title="E34">A27</font></i></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></i>;<br/><b>hence </b><a NAME="E12:15_1_1_2_4"/>
 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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E7:15_1_1_2_4"><i><font color="Green" title="E37">A30</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E16:15_1_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E5:15_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="#E7:15_1_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="#E11:15_1_1_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="#E13:15_1_1_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="#E14:15_1_1_2"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="qc_lang2.html#D20" title="QC_LANG2:def.20">QC_LANG2:def 20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c5">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> ) )
 <b>by </b><i><a class="txt" href="#E3:15_1_1"><i><font color="Green" title="E25">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E9:15_1"/>
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>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:15_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E8:15_1"><i><font color="Green" title="E22">A8</font></i></a>);<br/></i><b>then consider </b><font color="Maroon" title="c7">p1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E11:15_1"/><i><font color="Green" title="E23">A34</font></i>: 
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</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> <font color="Maroon" title="c6">L</font></span>)</span> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c5">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="c7">p1</font> &amp; ( <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c7">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="Maroon" title="c7">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="Maroon" title="c7">p1</font> implies <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c7">p1</font> ) )
 <b>by </b><i><a class="txt" href="#E6:15_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E12:15_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> <font color="Maroon" title="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E11:15_1"><i><font color="Green" title="E23">A34</font></i></a>, <a class="ref" href="substut2.html#D6" title="SUBSTUT2:def.6">SUBSTUT2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
( <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="c4">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
