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<b>let </b><font color="Maroon">A</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">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span>;<br/>







<b>set </b><font color="Maroon">g</font> =  <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font>;<br/>
<a NAME="E1:42"/><i><font color="Green">E29</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</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"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</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="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</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="E2:42"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:42"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:42"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:42"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> holds <br/><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </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="E1:42_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>

;<br/>
<b>hence </b><a NAME="E6:42_1"/>
<font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


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