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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">E1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">E2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="monoid_1.html#NM1">Function</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="monoid_1.html#NM1">Function</a> of <font color="Maroon">E1</font>,<font color="Maroon">E2</font>,<font color="Maroon">E</font>;<br/>







<b>assume </b><a NAME="E1:27"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">o1</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">E1</font>,<font color="Maroon">E2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E1</font></span>)</span>,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> holds <br/><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E3:27"/><b>then </b>
( <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E1</font> &amp; <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T138">ZFMISC_1:138</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E1</font> &amp;  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E2</font> )
 
<b>by </b><i><a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/>
<a NAME="E5:27"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D1</font> ;<br/><b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E1</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D2</font> ;<br/><b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">E2</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E7:27_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:27_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:27_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E4:27_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E9:27_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">o2</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">o1</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>hence </b><a NAME="E10:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon">o1</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">o2</font>,<font color="Maroon">E</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div>
