<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>







<a NAME="E1:15"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E2:15"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a></i>;<br/>
<a NAME="E3:15"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K18">Var1</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K19">Var2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T7">ZF_LANG:7</a></i>;<br/>
<a NAME="E4:15"/><i><font color="Green">E11</font></i>: 
 <a href="zf_model.html#K5" target="_self">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:15"><i><font color="Green">E9</font></i></a>, <a class="txt" href="zf_model.html#E7"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:15"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5" target="_self">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_model.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:15"><i><font color="Green">E10</font></i></a>, <a class="txt" href="zf_model.html#E1:15_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5" target="_self">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_model.html#E4:15"><i><font color="Green">E11</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5" target="_self">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:15"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="zf_model.html#E4:15"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_model.html#E3:15"><i><font color="Green">E10</font></i></a></i>;<br/>


</div>
