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

<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font>;<br/>
<b>take </b><font color="Maroon" title="c9">F</font> =  <a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c6">H</font>,<font color="Maroon" title="c8">f</font>;<br/>

<a NAME="E2:17_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:17_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c10">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font>;<br/>

<b>thus </b><a NAME="E4:17_1_1"/>
( <font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c10">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> implies <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">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" title="E14">A4</font></i>: 
<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c10">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">g3</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font><b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c11">g3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 holds <br/><font color="Maroon" title="c11">g3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21" title="ZF_MODEL:th.21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">g4</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font><b> such that </b><br/><a NAME="E5:17_1_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
( <font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 holds <br/><font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21" title="ZF_MODEL:th.21">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" title="c13">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:17_1_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 ;<br/><a NAME="E2:17_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:17_1_1_1"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/><a NAME="E3:17_1_1_1_1"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:17_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E5:17_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="zfmodel1.html#E1:17_1_1_1_1"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_1_1"/>
not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E17">A9</font></i>: 
<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c12">g4</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:17_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="zfmodel1.html#T14" target="_self" title="ZFMODEL1:th.14">Th14</a></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" title="c13">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_1_2"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E2:17_1_1_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> )
 ;<br/><font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c11">g3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:17_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_1_2"><i><font color="Green" title="E19">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:17_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_1_2"><i><font color="Green" title="E19">A11</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:17_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:17_1_1_1_2"><i><font color="Green" title="E18">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:17_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E2:17_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E3:17_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E7:17_1_1_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:17_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:17_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E5:17_1_1_1"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green" title="E14">A12</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>
 <b>and </b><br/><a NAME="E6:17_1_1"/><i><font color="Green" title="E15">A13</font></i>: 
not <font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c10">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">j</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font><b> such that </b><br/><a NAME="E8:17_1_1"/><i><font color="Green" title="E16">A14</font></i>: 
( <font color="Maroon" title="c11">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 holds <br/><font color="Maroon" title="c11">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21" title="ZF_MODEL:th.21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c7">E</font><b> such that </b><br/><a NAME="E10:17_1_1"/><i><font color="Green" title="E17">A15</font></i>: 
( <font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 holds <br/><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21" title="ZF_MODEL:th.21">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" title="c13">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 ;<br/><a NAME="E2:17_1_1_2"/><b>then </b><i><font color="Green" title="E19">A17</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:17_1_1"><i><font color="Green" title="E17">A15</font></i></a></i>;<br/><a NAME="E3:17_1_1_2"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E8:17_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E10:17_1_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="zfmodel1.html#E1:17_1_1_2"><i><font color="Green" title="E18">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_2"/>
not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:17_1_1"/><b>then </b><i><font color="Green" title="E18">A18</font></i>: 
not <font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c12">h</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E6:17_1_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="zfmodel1.html#T14" target="_self" title="ZFMODEL1:th.14">Th14</a></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" title="c13">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_3"/><i><font color="Green" title="E19">A19</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E2:17_1_1_3"/><i><font color="Green" title="E20">A20</font></i>: 
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font> )
 ;<br/><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c11">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E10:17_1_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_3"><i><font color="Green" title="E20">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E8:17_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E2:17_1_1_3"><i><font color="Green" title="E20">A20</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:17_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:17_1_1_3"><i><font color="Green" title="E19">A19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:17_1_1"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E2:17_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E3:17_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E12:17_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E15:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:17_1_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="zfmodel1.html#E8:17_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E10:17_1_1"><i><font color="Green" title="E17">A15</font></i></a></i>;<br/>


</div>
