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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">a<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">a<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> ];<br/>
<a NAME="E1:15_1"/><i><font color="Green">E40</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> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<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">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>



<b>consider </b><font color="Maroon">ai</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">bi</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:15_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font> <a href="domain_1.html#K2">`1</a> ,<font color="Maroon">x</font> <a href="domain_1.html#K3">`2</a> ,<font color="Maroon">ai</font>,<font color="Maroon">bi</font>]
 <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">ai</font>,<font color="Maroon">bi</font></span><a href="domain_1.html#K1">]</a></span>
;<br/>

<a NAME="E3:15_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">ai</font>,<font color="Maroon">bi</font></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ai</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">ai</font>,<font color="Maroon">bi</font></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">bi</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">ai</font>,<font color="Maroon">bi</font></span><a href="domain_1.html#K1">]</a></span>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">e</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i>
<b>take </b>
 <a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font>
;<br/>

<b>take </b>
 <a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font>
;<br/>

<a NAME="E5:15_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D6">FUNCT_2:def 6</a>, <a class="ref" href="funct_2.html#D7">FUNCT_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:15_1"/>
( <span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E7:15_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D6">FUNCT_2:def 6</a>, <a class="ref" href="funct_2.html#D7">FUNCT_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5">pr1</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i/>;<br/>


</div>
