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

<span class="kw">let </span><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">Z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">g</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font> holds <br/>( <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font> implies ( <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:66"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E2:66"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<a NAME="E3:66"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/>
<a NAME="E4:66"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T51" target="_self" onmouseover="rs('funct_3/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E5:66"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" target="_self" onmouseover="rs('funct_3/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E6:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E7:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_3.html#T50" target="_self" onmouseover="rs('funct_3/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E8:66"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E2:66_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:66_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:66_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="funct_3.html#D7" target="_self" onmouseover="rs('funct_3/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:66_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="funct_3.html#D4" target="_self" onmouseover="rs('funct_3/D4')" onmouseout="rh()">Def4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:66"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E10:66"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" target="_self" onmouseover="rs('funct_3/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E11:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_3.html#T50" target="_self" onmouseover="rs('funct_3/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E13:66"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_2"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E2:66_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A12</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E1:66_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:66"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E1:66_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="funct_3.html#D7" target="_self" onmouseover="rs('funct_3/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:66_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="funct_3.html#D5" target="_self" onmouseover="rs('funct_3/D5')" onmouseout="rh()">Def5</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E14:66"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
