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<div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">G</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">h</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">h</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Olive">h</font> holds <br/><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">q</font>
 ;<br/>

<b>assume </b><a NAME="E2:18_1_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">h</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">h</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Olive">h</font> holds <br/><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">q</font>
 ;<br/>

<b>thus </b><a NAME="E3:18_1_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_2_1"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E2:18_1_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">h</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">h</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">h</font> = 
<font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">h</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:18_1_2_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_2_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>


</div>
