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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><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">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">D</font>;<br/>









<b>assume </b><a NAME="E1:16"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:16"/><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K6" target="_self">[;]</a> <font color="Maroon">a</font>,<font color="Maroon">f</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">o</font> <a href="monoid_1.html#K7" target="_self">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</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="16_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="E1:16_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K6" target="_self">[;]</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T42">FUNCOP_1:42</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon">o</font> <a href="monoid_1.html#K6" target="_self">[;]</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><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="E1:16_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:16_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K7" target="_self">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T35">FUNCOP_1:35</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon">o</font> <a href="monoid_1.html#K7" target="_self">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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