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

<span class="kw">let </span><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b2">h</font>) holds <br/><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">q</font> ) &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b2">h</font>) holds <br/><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">q</font> ) implies <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b2">h</font>) holds <br/><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>) ex <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b2">h</font>) &amp;  not <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">q</font> ) or <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b2">h</font>) holds <br/><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font></span><br/>

<a NAME="E3:18_1_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c8">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<span class="kw"> such that </span><br/><a NAME="E2:18_1_2_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c8">h</font>) as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">h</font>) as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  ;<br/>
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">h</font>) = 
<font color="Maroon" title="c10">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c9">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">h</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:18_1_2_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:18_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
