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

<b>consider </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>;<br/>
<b>take </b><font color="Maroon">F</font> =  <a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font>;<br/>

<a NAME="E2:17_1_1"/><i><font color="Green">E38</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:17_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>

<b>thus </b><a NAME="E4:17_1_1"/>
( <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<b>consider </b><font color="Maroon">g3</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><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon">g4</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><br/><a NAME="E5:17_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:17_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:17_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_1_1"/>
 <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:17_1_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:17_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">g4</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:17_1_1_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/><font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E4:17_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E9:17_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:17_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>
 <b>and </b><br/><a NAME="E6:17_1_1"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<b>consider </b><font color="Maroon">j</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><br/><a NAME="E8:17_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon">h</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><br/><a NAME="E10:17_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:17_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_2"/>
 <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:17_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:17_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:17_1_1_3"/><i><font color="Green">E61</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E4:17_1_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E14:17_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E15:17_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
