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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">X1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">X2</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D2</font>;<br/>







<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> ) </span>}</span> ;<br/>
<a NAME="E1:75"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:75"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:75_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">X1</font>,<font color="Maroon">X2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:75_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:75"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:75_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="monoid_1.html#E3:75_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font> <b>by </b><i><a class="txt" href="monoid_1.html#E5:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> <b>by </b><i><a class="txt" href="monoid_1.html#E5:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E8:75_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
 
<b>by </b><i><a class="txt" href="monoid_1.html#E3:75_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="monoid_1.html#E5:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:75_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="monoid_1.html#E5:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:75"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> ) </span>}</span> 
 ;<br/>

<a NAME="E4:75"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font> ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
;<br/>
<b>hence </b><a NAME="E5:75"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">X1</font>,<font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


</div>
