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<b>let </b><font color="Maroon">X</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">Y</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">Z</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="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font>;<br/>









<b>assume </b><a NAME="E1:13"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> );<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1"/>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1"><b>case </b></a><a NAME="E1:13_1_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:13_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E4:13_1_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; (  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> implies  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2"><b>case </b></a><a NAME="E1:13_1_1_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E3:13_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>hence </b><a NAME="E4:13_1_1_1_2"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; (  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> implies  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:13_1"/><i><font color="Green">E34</font></i>: 
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> implies  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>thus </b><a NAME="E6:13_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">u</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">H</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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