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<b>let </b><font color="Maroon">E</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">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>given </b><font color="Maroon">H</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><a NAME="E1:19"/><i><font color="Green">E37</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">E</font> <a href="zf_model.html#R2">|=</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> &amp; <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K2" target="_self">def_func</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D3" target="_self">ZFMODEL1:def 3</a><br/>

<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>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D4" target="_self">ZFMODEL1:def 4</a><br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><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#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/><a NAME="E2:19_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:19_1"/>
not <font color="Maroon">a</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>
<b>hence </b><a NAME="E6:19"/>
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><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#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2"/>
 <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>
 ;<br/><a NAME="E2:19_2"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><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#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><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#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:19_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E8:19"/><i><font color="Green">E42</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/>

<a NAME="E9:19"/>
for <font color="Olive">g</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 ( for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</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="Olive">y</font> or  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R1">=</a> <font color="Olive">y</font> or  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#R1">=</a> <font color="Olive">y</font> or  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) ) holds <br/>( <font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> iff <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">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="Olive">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/>
<b>hence </b><a NAME="E10:19"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/>


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