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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">q</font> );<br/>
<b>let </b><font color="Maroon">p1</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">p2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27_1_2"/><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <font color="Maroon">f</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <font color="Maroon">f</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 ;<br/>

<b>consider </b><font color="Maroon">G</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:27_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E5:27_1_2"/><i><font color="Green">E58</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <font color="Maroon">f</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E7:27_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E8:27_1_2"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <font color="Maroon">f</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>by </b><i/>;<br/>
<a NAME="E9:27_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E10:27_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S15">RECDEF_1:sch 15</a>(, , );<br/></i>
<b>hence </b><a NAME="E11:27_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, </i>;<br/>


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