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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>



<a NAME="E1:25"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:25_1"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="margrel1.html#T49">MARGREL1:49</a></i>;<br/>
<a NAME="E2:25_1"/>
 <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green">E17</font></i>: 
(  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:25"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:25"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:25"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:25"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:25"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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