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

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">M</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">v</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/>





<b>set </b><font color="Maroon">S</font> =  <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> or  not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Olive">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:12_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:12_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:12_1_1_1_1"/>
<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="zf_model.html#D4">ZF_MODEL:def 4</a></i>;<br/>
<a NAME="E5:12_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D5">ZF_FUND1:def 5</a></i>;<br/>
<a NAME="E6:12_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:12_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:12_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:12_1_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> )
 ;<br/>
<a NAME="E4:12_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:12_1_1_1_2"/><b>then </b>
 ex <font color="Olive">v1</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v1</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">v1</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D5">ZF_FUND1:def 5</a></i>;<br/>
<a NAME="E6:12_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#T37">ZF_FUND1:37</a></i>;<br/>
<a NAME="E7:12_1_1_1_2"/><b>then </b>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#D4">ZF_MODEL:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:12_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_1_1_1"/>
 <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E52</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> ;<br/><a NAME="E3:12_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:12_1_1_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> )
 ;<br/><b>consider </b><font color="Maroon">v2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font><b> such that </b><br/><a NAME="E7:12_1_1_2_1"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="zf_fund1.html#D5">ZF_FUND1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><a NAME="E9:12_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:12_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:12_1_1_2_1"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">w</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E12:12_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">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="E13:12_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E15:12_1_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4">x".</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#T34">ZF_FUND1:34</a></i>;<br/><b>thus </b><a NAME="E16:12_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="zf_fund1.html#D3">ZF_FUND1:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12"/>
 <a href="zf_fund2.html#K1" target="_self">Section</a> <font color="Maroon">H</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">M</font> </span>}</span> 
 ;<br/>


</div>
