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<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">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>;<br/>
<a NAME="E1:46"/><i><font color="Green">E24</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#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</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="46_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="E1:46_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">t</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> <b>by </b><i><a class="txt" href="prob_4.html#E1:46_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E3:46_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">A</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:46"/>
 ex <font color="Olive">H</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</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">H</font> <a href="prob_1.html#K1">.</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#T1" target="_self">Th1</a>);<br/></i>
<b>then consider </b><font color="Maroon">H</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font><b> such that </b><br/><a NAME="E4:46"/><i><font color="Green">E39</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">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">t</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E5:46"/>
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">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i/>;<br/>


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