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<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#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  =  <a href="margrel1.html#K7">FALSE</a> ;<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#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">v</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp; <font color="Olive">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> );<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E25</font></i>: 
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>  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K6">BOOLEAN</a>  ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K6">BOOLEAN</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>, <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E30</font></i>: 
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>  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</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> <font color="Maroon">H<sub>2</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="Maroon">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>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>();<br/></i>
<a NAME="E4:16_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>

<a NAME="E6:16_1_1"/>
( (  ex <font color="Olive">v'</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">v</font> <a href="funct_2.html#R2">=</a> <font color="Olive">v'</font> &amp; <font color="Olive">v'</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> ) implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( ( for <font color="Olive">v'</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>  holds <br/>(  not <font color="Maroon">v</font> <a href="funct_2.html#R2">=</a> <font color="Olive">v'</font> or  not <font color="Olive">v'</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> ) ) implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:16_1_1"/>
( ( <font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( not <font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 ;<br/>


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