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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K1">VAR</a> ;<br/>



<a NAME="E1:14"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <font color="Maroon">c<sub>2</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E2:14"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:14"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="txt" href="zf_fund1.html#E1:14_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  <b>by </b><i><a class="txt" href="zf_fund1.html#E3:14_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<i><font color="Green">E21</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
<a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:14_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_fund1.html#E4"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
 <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D2" target="_self">Def2</a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E7:14_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:14_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_fund1.html#E6:14_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zf_fund1.html#E15"><i><font color="Green">Lemma14</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a> <b> such that </b><a NAME="E3:14"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E4:14"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" target="_self">decode</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:14"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_fund1.html#E17"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:14"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_fund1.html#E4"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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