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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> );<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:8_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<a NAME="E3:8_1_1"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><a NAME="E1:8_1_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> )
 ;<br/><a NAME="E2:8_1_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:8_1_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="ordinal1.html#K5">omega</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:8_1_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E5:8_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 ;<br/><a NAME="E6:8_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_1"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> ) )
 ;<br/>


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