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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:45"/>
<font color="Maroon">p</font> <a href="cqc_the3.html#R6" target="_self">is_an_universal_closure_of</a> <font color="Maroon">q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  ex <font color="Olive">L</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">r</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:45"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E6:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E7:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">r</font> )
 <b>by </b><i><a class="ref" href="cqc_the3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:45"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Olive">r</font> ) );<br/>
<a NAME="E1:45_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:45_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:45_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E2:45_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i><a class="ref" href="cqc_the3.html#D1" target="_self">Def1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_1"><b>case </b></a><a NAME="E1:45_1_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">p</font> = <font color="Maroon">q</font>;<br/><b>thus </b><a NAME="E2:45_1_1_1_1_1"/>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Olive">r</font> )
 <b>by </b><i><a class="txt" href="cqc_the3.html#E3:45"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_the3.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_2"><b>case </b></a><a NAME="E1:45_1_1_1_1_2"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">r</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:45_1_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:45_1_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:45_1_1_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/><a NAME="E7:45_1_1_1_1_2"/>
<font color="Maroon">s</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E8:45_1_1_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T4" target="_self">Th4</a>, <a class="ref" href="cqc_the3.html#T5" target="_self">Th5</a>, <a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a>, </i>;<br/><b>hence </b><a NAME="E9:45_1_1_1_1_2"/>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Olive">s</font> )
 <b>by </b><i><a class="ref" href="cqc_the3.html#D2" target="_self">Def2</a>, <a class="ref" href="cqc_the3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:45_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="cqc_the3.html#E6:45"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cqc_the3.html#E7:45"><i><font color="Green">E43</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E10:45"/><b>then </b>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Olive">r</font> )
 
<b>by </b><i><a class="ref" href="cqc_the3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E11:45"/>
<font color="Maroon">p</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T2" target="_self">Th2</a></i>;<br/>


</div>
