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<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> ;<br/><b>let </b><font color="Maroon">f</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">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>











<b>assume </b><a NAME="E1:13"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E44</font></i>: 
(  <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#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#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#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font> 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> <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f2</font> 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> <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:13"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</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">f</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</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> &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> <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;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</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> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>;<br/>
<b>hence </b><a NAME="E7:13"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="monoid_1.html#K1" target="_self">..</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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