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

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><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>assume </b><a NAME="E1:55"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1" target="_self">Cn</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:55"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:55"/><i><font color="Green">E30</font></i>: 
 <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:55"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:55"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a>  is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:55"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>;<br/>

<b>thus </b><a NAME="E10:55"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>

<b>thus </b><a NAME="E11:55"/>
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T15">FINSET_1:15</a></i>;<br/>

<a NAME="E12:55"/>
for <font color="Olive">n</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">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font>,<font color="Olive">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:55_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8 or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9 )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:55_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:55_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:55_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:55_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_3"><b>case </b></a><a NAME="E1:55_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_3"/>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_4"><b>case </b></a><a NAME="E1:55_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_4"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">q</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_5"><b>case </b></a><a NAME="E1:55_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_5"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font>, <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 <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6"><b>case </b></a><a NAME="E1:55_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_6"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">p</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_7"><b>case </b></a><a NAME="E1:55_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_7"/>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ex <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8"><b>case </b></a><a NAME="E1:55_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_8"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">q</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_9"><b>case </b></a><a NAME="E1:55_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_9"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ex <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">q</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">q</font></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_10"><b>case </b></a><a NAME="E1:55_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_10"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> ex <font color="Olive">s</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Olive">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Olive">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">y</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:55"/><b>then </b>
<font color="Maroon">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:55"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">Y</font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:55"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1" target="_self">Cn</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
