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

<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">BSeq</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>;<br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 ;<br/>

<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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E2:47"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>,<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:47_1"/>
<font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">t</font> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E2:47"><i><font color="Green">E36</font></i></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="E3:47_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="prob_4.html#E1:47"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E5:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">A</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47"/>
 ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">Sigma</font> st <br/>for <font color="Olive">t</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">t</font>,<font color="Olive">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="prob_4.html#T2" target="_self">Th2</a>);<br/></i>
<b>then consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">Sigma</font><b> such that </b><br/><a NAME="E5:47"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">t</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> holds <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<a NAME="E6:47"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">CSeq</font> = <font color="Maroon">G</font> as   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CSeq</font> = <font color="Maroon">CSeq</font> as    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">CSeq</font>
;<br/>

<b>thus </b><a NAME="E9:47"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">BSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">CSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div>
