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<b>let </b><font color="Maroon">F1</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>, <font color="Maroon">F2</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:85_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><br/> for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>  st <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font> holds <br/><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 <b>and </b><br/><a NAME="E2:85_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><br/> for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>  st <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font> holds <br/><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 ;<br/>

<a NAME="E3:85_1_2"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> holds  <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:85_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:85_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">d2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:85_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:85_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E5:85_1_2_1"/>
(  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font> &amp;  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> &amp; <font color="Maroon">c</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:85_1_2_1"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:85_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="cat_1.html#E2:85_1_2_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cat_1.html#E4:85_1_2_1"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:85_1_2_1"/>
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:85_1_2"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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