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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">p</font> be    <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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<a NAME="E1:50"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sigma</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) ) ) )
 
<b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:50"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sigma</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1 ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:50"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sigma</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">D</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E5:50"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 ;<br/>

<b>hence </b><a NAME="E6:50"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E7:50"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>hence </b><a NAME="E8:50"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="prob_1.html#T3" target="_self">Th3</a></i>;<br/>


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