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<b>let </b><font color="Maroon">f1</font> be    <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> , <font color="Maroon">f2</font> be    <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16_1_2"/><i><font color="Green">E32</font></i>: 
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/> ( ( <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">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( 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">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 <b>and </b><br/><a NAME="E2:16_1_2"/><i><font color="Green">E33</font></i>: 
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/> ( ( <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">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( 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">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 ;<br/>

<a NAME="E3:16_1_2"/>
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  <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_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> or  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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_1_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>
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">f1</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; <font color="Maroon">f2</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>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:16_1_2_1_1_1"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_2"/>
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>
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">f1</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>  &amp; <font color="Maroon">f2</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>, </i>;<br/><b>hence </b><a NAME="E3:16_1_2_1_1_2"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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