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

<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> ;<br/><b>let </b><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="E1:11_1_2"/><i><font color="Green">E30</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  <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#K12">ALL</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v'</font></span>)</span> where <font color="Olive">v'</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> : for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">v'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 <b>and </b><br/><a NAME="E2:11_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  <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#K12">ALL</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v'</font></span>)</span> where <font color="Olive">v'</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> : for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">v'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/>

<a NAME="E3:11_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="11_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/>

<a NAME="E1:11_1_2_1"/>
( <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#K12">ALL</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v'</font></span>)</span> where <font color="Olive">v'</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> : for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">v'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span>  &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#K12">ALL</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v''</font></span>)</span> where <font color="Olive">v''</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> : for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">v''</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:11_1_2_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>
<b>hence </b><a NAME="E4:11_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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