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

<b>let </b><font color="Maroon" title="c6">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> holds <br/><font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a><br/>

<div><i><font color="Green" title="E15">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Olive" title="b1">H</font>
 ;<br/><b>let </b><font color="Maroon" title="c7">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c8">f</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="c6">E</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:20_1"/><i><font color="Green" title="E17">A4</font></i>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font>
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c6">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="c7">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>
 ;<br/><a NAME="E4:20_1"/>
 <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> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E5:20_1"/><b>then </b><i><font color="Green" title="E19">A6</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="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:20_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1"/>
not <font color="Maroon" title="c9">a</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="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E7:20_1"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font>;<br/><a NAME="E8:20_1"/>
<font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/><a NAME="E9:20_1"/><b>then </b>
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c7">H</font> &amp;  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p3">(<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="p3">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a></i>;<br/><a NAME="E10:20_1"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c6">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> 1</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> 2</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"><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#K5" title="ZF_LANG:func.5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="zf_model.html#T18" title="ZF_MODEL:th.18">ZF_MODEL:18</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">g</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="c6">E</font><b> such that </b><br/><a NAME="E12:20_1"/><i><font color="Green" title="E22">A9</font></i>: 
( <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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u</font> &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> 1 holds <br/><font color="Maroon" title="c10">g</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/><a NAME="E13:20_1"/>
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="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/><a NAME="E14:20_1"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c10">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:20_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">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="c6">E</font><b> such that </b><br/><a NAME="E16:20_1"/><i><font color="Green" title="E23">A10</font></i>: 
( ( 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="Maroon" title="c11">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#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="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c11">h</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> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p3">(<span class="default"><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> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/><b>set </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">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> 2</span>)</span>;<br/><a NAME="E17:20_1"/><i><font color="Green" title="E24">A11</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1_2"/><i><font color="Green" title="E26">A13</font></i>: 
( <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">u</font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">e</font> = <font color="Maroon" title="c13">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2"><i><font color="Green" title="E25">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green" title="E26">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">i</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="c6">E</font><b> such that </b><br/><a NAME="E7:20_1_2"/><i><font color="Green" title="E27">A14</font></i>: 
( <font color="Maroon" title="c17">i</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="c15">e</font> &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="c17">i</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">h</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/>
<a NAME="E8:20_1_2"/>
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="Maroon" title="c17">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a></i>;<br/>
<a NAME="E9:20_1_2"/><b>then </b><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <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> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">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="c6">E</font><b> such that </b><br/><a NAME="E11:20_1_2"/><i><font color="Green" title="E29">A16</font></i>: 
( <font color="Maroon" title="c18">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="c16">b</font> &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="c18">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="c17">i</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/>
<a NAME="E12:20_1_2"/>
( <font color="Maroon" title="c18">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</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> 1</span>)</span> &amp; <font color="Maroon" title="c17">i</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 1</span>)</span> &amp; <font color="Maroon" title="c11">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> 1</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> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/>
<a NAME="E13:20_1_2"/><b>then </b><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c18">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">m1</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="c6">E</font><b> such that </b><br/><a NAME="E15:20_1_2"/><i><font color="Green" title="E31">A18</font></i>: 
( <font color="Maroon" title="c19">m1</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="c16">b</font> &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="c19">m1</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="c20">m</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="c6">E</font><b> such that </b><br/><a NAME="E17:20_1_2"/><i><font color="Green" title="E32">A19</font></i>: 
( <font color="Maroon" title="c20">m</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="c15">e</font> &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="c20">m</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="c19">m1</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="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:20_1_2_1"/><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c20">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</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="c21">y</font>
 ;<br/><b>assume </b><a NAME="E2:20_1_2_1"/>
(  <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="c21">y</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="c21">y</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="c21">y</font> )
 ;<br/><a NAME="E3:20_1_2_1"/><b>then </b>
( <font color="Maroon" title="c20">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</font> &amp; <font color="Maroon" title="c19">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</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="Maroon" title="c21">y</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2_1"><i><font color="Green" title="E33">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:20_1_2"/><b>then </b><i><font color="Green" title="E33">A21</font></i>: 
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c20">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</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="c20">m</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#E3:20_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/>
<a NAME="E20:20_1_2"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c20">m</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="c19">m1</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>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/>
<a NAME="E21:20_1_2"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c20">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="zfmodel1.html#E19:20_1_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self" title="ZFMODEL1:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">k</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="c6">E</font><b> such that </b><br/><a NAME="E23:20_1_2"/><i><font color="Green" title="E36">A24</font></i>: 
( <font color="Maroon" title="c21">k</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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> 1</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> 1 holds <br/><font color="Maroon" title="c21">k</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="c20">m</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/>
<a NAME="E24:20_1_2"/>
(  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font></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="Maroon" title="c21">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/>
<a NAME="E25:20_1_2"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c21">k</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> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E21:20_1_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">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="E1:20_1_2_2"/><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
 <b>and </b><br/><a NAME="E2:20_1_2_2"/><i><font color="Green" title="E39">A27</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">x</font>
 ;<br/><a NAME="E3:20_1_2_2"/><i><font color="Green" title="E40">A28</font></i>: 
<font color="Maroon" title="c22">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> 1
 <b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green" title="E38">A26</font></i></a></i>;<br/><a NAME="E4:20_1_2_2"/><i><font color="Green" title="E41">A29</font></i>: 
<font color="Maroon" title="c22">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
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_1_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green" title="E38">A26</font></i></a></i>;<br/><a NAME="E5:20_1_2_2"/>
( <font color="Maroon" title="c21">k</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="c20">m</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; <font color="Maroon" title="c18">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> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</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#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/><a NAME="E6:20_1_2_2"/><b>then </b><i><font color="Green" title="E42">A30</font></i>: 
<font color="Maroon" title="c22">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
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green" title="E38">A26</font></i></a></i>;<br/><font color="Maroon" title="c21">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> = 
<font color="Maroon" title="c20">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2_2"><i><font color="Green" title="E40">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="zfmodel1.html#E6:20_1_2_2"><i><font color="Green" title="E42">A30</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="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_1_2_2"><i><font color="Green" title="E41">A29</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2_2"><i><font color="Green" title="E40">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1_2_2"><i><font color="Green" title="E39">A27</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="zfmodel1.html#E6:20_1_2_2"><i><font color="Green" title="E42">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_1_2_2"><i><font color="Green" title="E41">A29</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:20_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green" title="E38">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:20_1_2"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c18">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E25:20_1_2"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E28:20_1_2"/><b>then </b><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c18">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_1_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E29:20_1_2"/>
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="Maroon" title="c17">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/>
<a NAME="E30:20_1_2"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E28:20_1_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/>
<a NAME="E31:20_1_2"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="zf_model.html#T19" title="ZF_MODEL:th.19">ZF_MODEL:19</a></i>;<br/>
<a NAME="E32:20_1_2"/><b>then </b>
( <font color="Maroon" title="c17">i</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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">i</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> 2</span>)</span> &amp; <font color="Maroon" title="c17">i</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E33:20_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green" title="E27">A14</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:20_1"/>
<font color="Maroon" title="c11">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> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_3"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 ;<br/>

<a NAME="E2:20_1_3"/>
<font color="Maroon" title="c11">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> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3"><i><font color="Green" title="E25">A32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">i</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="c6">E</font><b> such that </b><br/><a NAME="E5:20_1_3"/><i><font color="Green" title="E26">A33</font></i>: 
( <font color="Maroon" title="c15">i</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="c14">e</font> &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="c15">i</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">h</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/>
<a NAME="E6:20_1_3"/>
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="Maroon" title="c15">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/>
<a NAME="E7:20_1_3"/><b>then </b><i><font color="Green" title="E27">A34</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <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> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E8:20_1_3"/>
<font color="Maroon" title="c15">i</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/>
<a NAME="E9:20_1_3"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E10:20_1_3"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_3"><i><font color="Green" title="E27">A34</font></i></a>, <a class="ref" href="zf_model.html#T19" title="ZF_MODEL:th.19">ZF_MODEL:19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">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="c6">E</font><b> such that </b><br/><a NAME="E12:20_1_3"/><i><font color="Green" title="E28">A35</font></i>: 
( ( 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="Maroon" title="c16">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c16">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/>
<a NAME="E13:20_1_3"/><i><font color="Green" title="E29">A36</font></i>: 
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c16">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c16">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E14:20_1_3"/><b>then </b><i><font color="Green" title="E30">A37</font></i>: 
( <font color="Maroon" title="c16">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">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> 1</span>)</span> &amp; <font color="Maroon" title="c16">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</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> 1</span>)</span> &amp; <font color="Maroon" title="c15">i</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 1</span>)</span> &amp; <font color="Maroon" title="c11">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> 1</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> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">m1</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="c6">E</font><b> such that </b><br/><a NAME="E16:20_1_3"/><i><font color="Green" title="E31">A38</font></i>: 
( <font color="Maroon" title="c17">m1</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="c16">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> &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="c17">m1</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="c18">m</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="c6">E</font><b> such that </b><br/><a NAME="E18:20_1_3"/><i><font color="Green" title="E32">A39</font></i>: 
( <font color="Maroon" title="c18">m</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="c14">e</font> &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="c18">m</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="c17">m1</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="20_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:20_1_3_1"/><i><font color="Green" title="E33">A40</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">y</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="c19">y</font>
 ;<br/><b>assume </b><a NAME="E2:20_1_3_1"/>
(  <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="c19">y</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="c19">y</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="c19">y</font> )
 ;<br/><a NAME="E3:20_1_3_1"/><b>then </b>
( <font color="Maroon" title="c18">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c17">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">y</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="Maroon" title="c19">y</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green" title="E32">A39</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3_1"><i><font color="Green" title="E33">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:20_1_3"/><b>then </b><i><font color="Green" title="E33">A41</font></i>: 
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c18">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</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="c18">m</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#E3:20_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/>
<a NAME="E21:20_1_3"/><i><font color="Green" title="E34">A42</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c16">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_1_3"><i><font color="Green" title="E29">A36</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self" title="ZFMODEL1:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">k</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="c6">E</font><b> such that </b><br/><a NAME="E23:20_1_3"/><i><font color="Green" title="E35">A43</font></i>: 
( <font color="Maroon" title="c19">k</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m</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> 1</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> 1 holds <br/><font color="Maroon" title="c19">k</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="c16">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/>
<a NAME="E24:20_1_3"/>
(  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font></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="Maroon" title="c19">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green" title="E35">A43</font></i></a></i>;<br/>
<a NAME="E25:20_1_3"/><b>then </b><i><font color="Green" title="E36">A44</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c19">k</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> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E21:20_1_3"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">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="E1:20_1_3_2"/><i><font color="Green" title="E37">A45</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
 <b>and </b><br/><a NAME="E2:20_1_3_2"/><i><font color="Green" title="E38">A46</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font>
 ;<br/><a NAME="E3:20_1_3_2"/><i><font color="Green" title="E39">A47</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green" title="E35">A43</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green" title="E37">A45</font></i></a></i>;<br/><a NAME="E4:20_1_3_2"/>
( <font color="Maroon" title="c19">k</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="c16">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> &amp; <font color="Maroon" title="c17">m1</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="c18">m</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> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green" title="E35">A43</font></i></a></i>;<br/><a NAME="E5:20_1_3_2"/><b>then </b><i><font color="Green" title="E40">A48</font></i>: 
 <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="c20">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green" title="E37">A45</font></i></a></i>;<br/><a NAME="E6:20_1_3_2"/>
( <font color="Maroon" title="c19">k</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="c16">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> 4</span>)</span> &amp; <font color="Maroon" title="c16">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> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</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#E12:20_1_3"><i><font color="Green" title="E28">A35</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green" title="E35">A43</font></i></a></i>;<br/><a NAME="E7:20_1_3_2"/><b>then </b><i><font color="Green" title="E41">A49</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="c20">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green" title="E37">A45</font></i></a></i>;<br/><font color="Maroon" title="c19">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font> = 
<font color="Maroon" title="c16">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green" title="E35">A43</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_3_2"><i><font color="Green" title="E39">A47</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green" title="E28">A35</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3_2"><i><font color="Green" title="E40">A48</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_3_2"><i><font color="Green" title="E41">A49</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1_3_2"><i><font color="Green" title="E38">A46</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="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_3_2"><i><font color="Green" title="E39">A47</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3_2"><i><font color="Green" title="E40">A48</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_3_2"><i><font color="Green" title="E41">A49</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green" title="E37">A45</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:20_1_3"/><b>then </b><i><font color="Green" title="E37">A50</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</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="c13">a</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_1_3"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="zfmodel1.html#E25:20_1_3"><i><font color="Green" title="E36">A44</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E28:20_1_3"/>
( <font color="Maroon" title="c16">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E29:20_1_3"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="zfmodel1.html#E14:20_1_3"><i><font color="Green" title="E30">A37</font></i></a>, <a class="txt" href="zfmodel1.html#E27:20_1_3"><i><font color="Green" title="E37">A50</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:20_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E20:20_1"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E16">A51</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">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="c6">E</font>  st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">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="Olive" title="b1">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> holds <br/>for <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> holds  <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font>
 ;<br/><b>let </b><font color="Maroon" title="c7">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>assume </b><a NAME="E2:20_2"/><i><font color="Green" title="E17">A52</font></i>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font>
 ;<br/><a NAME="E3:20_2"/>
 <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> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E4:20_2"/><b>then </b><i><font color="Green" title="E18">A53</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="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green" title="E17">A52</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:20_2_1"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:20_2_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:20_2_1"/>
not <font color="Maroon" title="c8">a</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="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green" title="E17">A52</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E6:20_2"/><b>then </b><i><font color="Green" title="E19">A54</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E7:20_2"/>
<font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c7">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</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="c6">E</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1"/><i><font color="Green" title="E20">A55</font></i>: 
<font color="Maroon" title="c6">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="c7">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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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="c6">E</font>;<br/><b>assume </b><a NAME="E1:20_2_2_1_1"/><i><font color="Green" title="E21">A56</font></i>: 
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="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">v</font> = <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2"><i><font color="Green" title="E16">A51</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green" title="E17">A52</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green" title="E20">A55</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">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="c6">E</font><b> such that </b><br/><a NAME="E4:20_2_2_1_1"/><i><font color="Green" title="E22">A57</font></i>: 
( <font color="Maroon" title="c11">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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> &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> 2 holds <br/><font color="Maroon" title="c11">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="c9">g</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="20_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">i</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="c6">E</font>;<br/><b>assume </b><a NAME="E1:20_2_2_1_1_1"/><i><font color="Green" title="E23">A58</font></i>: 
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="Maroon" title="c12">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/><div><i><font color="Green" title="E24">A59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1_1_1_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c12">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>
 ;<br/><a NAME="E2:20_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">i</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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">i</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> 2</span>)</span> &amp; <font color="Maroon" title="c12">i</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_2_2_1_1_1_1"/><i><font color="Green" title="E25">A60</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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> 1</span>)</span> &amp; <font color="Maroon" title="c12">i</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> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c13">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green" title="E25">A60</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">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="c6">E</font><b> such that </b><br/><a NAME="E7:20_2_2_1_1_1_1"/><i><font color="Green" title="E26">A61</font></i>: 
( <font color="Maroon" title="c15">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="c14">a</font> &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="c15">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="c12">i</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/><a NAME="E8:20_2_2_1_1_1_1"/>
( <font color="Maroon" title="c15">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</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> 1</span>)</span> &amp; <font color="Maroon" title="c12">i</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 1</span>)</span> &amp; <font color="Maroon" title="c11">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a></i>;<br/><a NAME="E9:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E27">A62</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green" title="E25">A60</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">m1</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="c6">E</font><b> such that </b><br/><a NAME="E11:20_2_2_1_1_1_1"/><i><font color="Green" title="E28">A63</font></i>: 
( <font color="Maroon" title="c16">m1</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="c15">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> &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="c16">m1</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="c17">m</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="c6">E</font><b> such that </b><br/><a NAME="E13:20_2_2_1_1_1_1"/><i><font color="Green" title="E29">A64</font></i>: 
( <font color="Maroon" title="c17">m</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="c12">i</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="c17">m</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="c16">m1</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="20_2_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">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="E1:20_2_2_1_1_1_1_1"/><i><font color="Green" title="E30">A65</font></i>: 
<font color="Maroon" title="c17">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">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="c18">x</font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_1_1"/><i><font color="Green" title="E31">A66</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="c18">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="c18">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="c18">x</font> )
 ;<br/><a NAME="E3:20_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c17">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c16">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">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="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green" title="E28">A63</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_1_1"><i><font color="Green" title="E31">A66</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_1"><i><font color="Green" title="E30">A65</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:20_2_2_1_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</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="c17">m</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; ( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</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="c17">m</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> implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> ) &amp; <font color="Maroon" title="c17">m</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="c16">m1</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> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2"><i><font color="Green" title="E18">A53</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green" title="E20">A55</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/><a NAME="E16:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E30">A67</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c17">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E6:20_2"><i><font color="Green" title="E19">A54</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green" title="E25">A60</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green" title="E28">A63</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self" title="ZFMODEL1:th.11">Th11</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">k</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="c6">E</font><b> such that </b><br/><a NAME="E18:20_2_2_1_1_1_1"/><i><font color="Green" title="E31">A68</font></i>: 
( <font color="Maroon" title="c18">k</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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> 1</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> 1 holds <br/><font color="Maroon" title="c18">k</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="c17">m</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/><a NAME="E19:20_2_2_1_1_1_1"/>
(  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font></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="Maroon" title="c18">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green" title="E31">A68</font></i></a></i>;<br/><a NAME="E20:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E32">A69</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c18">k</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> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_2_2_1_1_1_1"><i><font color="Green" title="E30">A67</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">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="E1:20_2_2_1_1_1_1_2"/><i><font color="Green" title="E33">A70</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_1_2"/><i><font color="Green" title="E34">A71</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">x</font>
 ;<br/><a NAME="E3:20_2_2_1_1_1_1_2"/><i><font color="Green" title="E35">A72</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green" title="E31">A68</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green" title="E33">A70</font></i></a></i>;<br/><a NAME="E4:20_2_2_1_1_1_1_2"/>
( <font color="Maroon" title="c18">k</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="c17">m</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; <font color="Maroon" title="c17">m</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="c16">m1</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> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green" title="E31">A68</font></i></a></i>;<br/><a NAME="E5:20_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A73</font></i>: 
 <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="c19">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green" title="E28">A63</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green" title="E33">A70</font></i></a></i>;<br/><a NAME="E6:20_2_2_1_1_1_1_2"/>
( <font color="Maroon" title="c15">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> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</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; <font color="Maroon" title="c18">k</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="c17">m</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#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green" title="E31">A68</font></i></a></i>;<br/><a NAME="E7:20_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A74</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="c19">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green" title="E33">A70</font></i></a></i>;<br/><font color="Maroon" title="c15">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> = 
<font color="Maroon" title="c12">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_1_2"><i><font color="Green" title="E36">A73</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1_2"><i><font color="Green" title="E37">A74</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_1_2"><i><font color="Green" title="E34">A71</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="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1"><i><font color="Green" title="E21">A56</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_1_2"><i><font color="Green" title="E35">A72</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green" title="E28">A63</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_1_2"><i><font color="Green" title="E36">A73</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green" title="E29">A64</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1_2"><i><font color="Green" title="E37">A74</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green" title="E31">A68</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_1_2"><i><font color="Green" title="E35">A72</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green" title="E33">A70</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E22:20_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E20:20_2_2_1_1_1_1"><i><font color="Green" title="E32">A69</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><a NAME="E23:20_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font> &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="Maroon" title="c15">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green" title="E26">A61</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_1"><i><font color="Green" title="E27">A62</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/><b>hence </b><a NAME="E24:20_2_2_1_1_1_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c12">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1_1_1_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c12">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">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="c6">E</font><b> such that </b><br/><a NAME="E3:20_2_2_1_1_1_2"/><i><font color="Green" title="E25">A75</font></i>: 
( ( 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="Maroon" title="c13">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c13">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/><a NAME="E4:20_2_2_1_1_1_2"/><i><font color="Green" title="E26">A76</font></i>: 
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c13">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c13">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green" title="E25">A75</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/><a NAME="E5:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A77</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c13">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E6:20_2"><i><font color="Green" title="E19">A54</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self" title="ZFMODEL1:th.11">Th11</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">m1</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="c6">E</font><b> such that </b><br/><a NAME="E7:20_2_2_1_1_1_2"/><i><font color="Green" title="E28">A78</font></i>: 
( <font color="Maroon" title="c14">m1</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="c13">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> &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="c14">m1</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="c15">m</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="c6">E</font><b> such that </b><br/><a NAME="E9:20_2_2_1_1_1_2"/><i><font color="Green" title="E29">A79</font></i>: 
( <font color="Maroon" title="c15">m</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="c12">i</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="c15">m</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="c14">m1</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="c16">k</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="c6">E</font><b> such that </b><br/><a NAME="E11:20_2_2_1_1_1_2"/><i><font color="Green" title="E30">A80</font></i>: 
( <font color="Maroon" title="c16">k</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</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> 1</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> 1 holds <br/><font color="Maroon" title="c16">k</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="c13">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/><a NAME="E12:20_2_2_1_1_1_2"/>
(  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<font color="Maroon" title="c7">H</font></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="Maroon" title="c16">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green" title="E30">A80</font></i></a></i>;<br/><a NAME="E13:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A81</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c16">k</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> 2</span>)</span>,<font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2"><i><font color="Green" title="E27">A77</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">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="E1:20_2_2_1_1_1_2_1"/><i><font color="Green" title="E32">A82</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_2_1"/><i><font color="Green" title="E33">A83</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">x</font>
 ;<br/><a NAME="E3:20_2_2_1_1_1_2_1"/><i><font color="Green" title="E34">A84</font></i>: 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green" title="E30">A80</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green" title="E32">A82</font></i></a></i>;<br/><a NAME="E4:20_2_2_1_1_1_2_1"/>
( <font color="Maroon" title="c15">m</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="c14">m1</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; <font color="Maroon" title="c16">k</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="c13">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> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green" title="E30">A80</font></i></a></i>;<br/><a NAME="E5:20_2_2_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E35">A85</font></i>: 
 <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="c17">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green" title="E28">A78</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green" title="E32">A82</font></i></a></i>;<br/><a NAME="E6:20_2_2_1_1_1_2_1"/>
( <font color="Maroon" title="c16">k</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="c13">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> 4</span>)</span> &amp; <font color="Maroon" title="c13">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> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</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#E3:20_2_2_1_1_1_2"><i><font color="Green" title="E25">A75</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green" title="E30">A80</font></i></a></i>;<br/><a NAME="E7:20_2_2_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E36">A86</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="c17">x</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green" title="E32">A82</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c15">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c14">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</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="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green" title="E28">A78</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2_1"><i><font color="Green" title="E35">A85</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1"><i><font color="Green" title="E21">A56</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2_1"><i><font color="Green" title="E34">A84</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_1"><i><font color="Green" title="E33">A83</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2_1"><i><font color="Green" title="E36">A86</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green" title="E25">A75</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2_1"><i><font color="Green" title="E35">A85</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green" title="E30">A80</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2_1"><i><font color="Green" title="E34">A84</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_2_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green" title="E32">A82</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E32">A87</font></i>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c15">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_2"><i><font color="Green" title="E31">A81</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">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="E1:20_2_2_1_1_1_2_2"/><i><font color="Green" title="E33">A88</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">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="c17">x</font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_2_2"/><i><font color="Green" title="E34">A89</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="c17">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="c17">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="c17">x</font> )
 ;<br/><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c14">m1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green" title="E28">A78</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_2"><i><font color="Green" title="E34">A89</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_2"><i><font color="Green" title="E34">A89</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:20_2_2_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_2"><i><font color="Green" title="E33">A88</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E33">A90</font></i>: 
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</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="c15">m</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; <font color="Maroon" title="c15">m</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="c14">m1</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> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2"><i><font color="Green" title="E18">A53</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green" title="E20">A55</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_2_2_1_1_1_2"><i><font color="Green" title="E32">A87</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self" title="ZFMODEL1:def.1">Def1</a></i>;<br/><a NAME="E18:20_2_2_1_1_1_2"/><i><font color="Green" title="E34">A91</font></i>: 
( <font color="Maroon" title="c13">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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> 1</span>)</span> &amp; <font color="Maroon" title="c13">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</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> 1</span>)</span> &amp; <font color="Maroon" title="c12">i</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 1</span>)</span> &amp; <font color="Maroon" title="c11">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green" title="E25">A75</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_2"><i><font color="Green" title="E26">A76</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><a NAME="E19:20_2_2_1_1_1_2"/>
( <font color="Maroon" title="c13">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E35">A92</font></i>: 
<font color="Maroon" title="c12">i</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#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c7">H</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green" title="E28">A78</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green" title="E29">A79</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_2_2_1_1_1_2"><i><font color="Green" title="E33">A90</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_2"><i><font color="Green" title="E34">A91</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E21:20_2_2_1_1_1_2"/>
<font color="Maroon" title="c12">i</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green" title="E23">A58</font></i></a></i>;<br/><b>hence </b><a NAME="E22:20_2_2_1_1_1_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c12">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</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="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_2_2_1_1_1_2"><i><font color="Green" title="E35">A92</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:20_2_2_1_1_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c12">i</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <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> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1"><i><font color="Green" title="E24">A59</font></i></a>, <a class="ref" href="zf_model.html#T19" title="ZF_MODEL:th.19">ZF_MODEL:19</a></i>;<br/></div><b>end;</b></div><a NAME="E6:20_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c11">h</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> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p3">(<span class="default"><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> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></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="Maroon" title="c11">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green" title="E22">A57</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E7:20_2_2_1_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_2_2_1"/>
<font color="Maroon" title="c6">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> 1</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> 2</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"><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#K5" title="ZF_LANG:func.5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_2_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T18" title="ZF_MODEL:th.18">ZF_MODEL:18</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
( ( for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Olive" title="b1">H</font> ) iff for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">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="c6">E</font>  st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><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#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">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="Olive" title="b1">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> holds <br/>for <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> holds  <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>


</div>
