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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">o</font> be   <a href="binop_1.html#NM2">BinOp</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>assume </b><a NAME="E1:26"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice2.html#D1">LATTICE2:def 1</a><br/>

<b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice2.html#D1">LATTICE2:def 1</a><br/>

<a NAME="E2:26"/><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">o'</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">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">o</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">f</font>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:26"/><i><font color="Green">E45</font></i>: 
(  <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#K4">dom</a> <font color="Maroon">g</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">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</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#K3" target="_self">.:</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></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="26_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="E1:26_1"/><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/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">g</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> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:26_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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#T28">FUNCOP_1:28</a></i>;<br/><b>hence </b><a NAME="E4:26_1"/>
<font color="Maroon">f</font> <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">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">o'</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></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="E5:26"/>
<span class="p1">(<span class="default"><font color="Maroon">o</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">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o'</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">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</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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