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

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><br/> for <font color="Olive" title="b3">E</font> being 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/> for <font color="Olive" title="b4">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b5">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="Olive" title="b3">E</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b4">H</font> &amp; <font color="Olive" title="b3">E</font>,<font color="Olive" title="b5">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b4">H</font> holds <br/><font color="Olive" title="b3">E</font>,<font color="Olive" title="b5">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> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">H</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>let </b><font color="Maroon" title="c8">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>let </b><font color="Maroon" title="c9">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="c10">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="c8">E</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E2:10"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E8">A3</font></i>: 
( not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font> )
 ;<br/>

<div><i><font color="Green" title="E9">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon" title="c9">H</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a> 
 ;<br/><a NAME="E2:10_1"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c9">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_1"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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="c8">E</font>;<br/><b>assume </b><a NAME="E1:10_1_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 ;<br/><a NAME="E2:10_1_1"/><b>then </b>
( <font color="Maroon" title="c11">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; <font color="Maroon" title="c11">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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="zfmodel1.html#E3:10_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <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>
<div><i><font color="Green" title="E10">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon" title="c9">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 ;<br/><a NAME="E2:10_2"/><b>then </b><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c9">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_2"/><b>then </b><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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="c8">E</font>;<br/><b>assume </b><a NAME="E1:10_2_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 ;<br/><a NAME="E2:10_2_1"/><b>then </b>
( <font color="Maroon" title="c11">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; <font color="Maroon" title="c11">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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10_2"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="zfmodel1.html#E3:10_2"><i><font color="Green" title="E12">A9</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="E5:10_2"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <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>
<div><i><font color="Green" title="E11">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_3"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 ;<br/><a NAME="E2:10_3"/><b>then </b><i><font color="Green" title="E13">A12</font></i>: 
( <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><b>assume </b><a NAME="E3:10_3"/>
not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">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="c8">E</font><b> such that </b><br/><a NAME="E5:10_3"/><i><font color="Green" title="E14">A13</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 <b>and </b><br/><a NAME="E6:10_3"/><i><font color="Green" title="E15">A14</font></i>: 
not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><a NAME="E7:10_3"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="zfmodel1.html#E6:10_3"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/><a NAME="E8:10_3"/>
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="zf_lang.html#D39" title="ZF_LANG:def.39">ZF_LANG:def 39</a></i>;<br/><a NAME="E9:10_3"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E1:10_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="zf_lang.html#T81" title="ZF_LANG:th.81">ZF_LANG:81</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><a NAME="E10:10_3"/><b>then </b>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</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> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E7:10_3"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/><a NAME="E11:10_3"/><b>then </b>
( <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font> &amp; not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="zfmodel1.html#E5:10_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</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="E12:10_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E12">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_4"/>
<font color="Maroon" title="c9">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 ;<br/><a NAME="E2:10_4"/><b>then </b><i><font color="Green" title="E13">A17</font></i>: 
( <font color="Maroon" title="c9">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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T58" title="ZF_LANG:th.58">ZF_LANG:58</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_4"/><b>then </b><i><font color="Green" title="E14">A18</font></i>: 
( <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/><a NAME="E4:10_4"/>
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_4"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="zf_lang.html#D39" title="ZF_LANG:def.39">ZF_LANG:def 39</a></i>;<br/><a NAME="E5:10_4"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T81" title="ZF_LANG:th.81">ZF_LANG:81</a></i>;<br/><a NAME="E6:10_4"/><b>then </b><i><font color="Green" title="E15">A19</font></i>: 
( <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E3:10_4"><i><font color="Green" title="E14">A18</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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="c8">E</font>;<br/><b>assume </b><a NAME="E1:10_4_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 ;<br/><a NAME="E2:10_4_1"/><b>then </b>
( <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E6:10_4"><i><font color="Green" title="E15">A19</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="E3:10_4_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_4"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:10_4"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5"/><i><font color="Green" title="E13">A20</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 ;<br/><a NAME="E2:10_5"/><b>then </b><i><font color="Green" title="E14">A21</font></i>: 
( <font color="Maroon" title="c9">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#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>;<br/><div><i><font color="Green" title="E15">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5_1"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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="c8">E</font>;<br/><b>assume </b><a NAME="E1:10_5_1_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  holds <br/>(  not <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> or  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 ;<br/><a NAME="E2:10_5_1_1"/><b>then </b>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10_5_1"><i><font color="Green" title="E16">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_5_1_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10_5"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div><a NAME="E3:10_5_1"/><b>then </b>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">f</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> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T22" title="ZF_MODEL:th.22">ZF_MODEL:22</a></i>;<br/><b>hence </b><a NAME="E4:10_5_1"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10_5"><i><font color="Green" title="E13">A20</font></i></a>, <a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></i>;<br/></div><b>end;</b></div><a NAME="E4:10_5"/><i><font color="Green" title="E16">A24</font></i>: 
(  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10_5"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5_2"/><i><font color="Green" title="E17">A25</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font>
 ;<br/><b>assume </b><a NAME="E2:10_5_2"/>
not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">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="c8">E</font><b> such that </b><br/><a NAME="E4:10_5_2"/><i><font color="Green" title="E18">A26</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 <b>and </b><br/><a NAME="E5:10_5_2"/><i><font color="Green" title="E19">A27</font></i>: 
not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c11">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><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="c12">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c8">E</font><b> such that </b><br/><a NAME="E7:10_5_2"/><i><font color="Green" title="E20">A28</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 <b>and </b><br/><a NAME="E8:10_5_2"/><i><font color="Green" title="E21">A29</font></i>: 
not <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c12">h</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_5"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="zfmodel1.html#E5:10_5_2"><i><font color="Green" title="E19">A27</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="c13">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="c8">E</font><b> such that </b><br/><a NAME="E10:10_5_2"/><i><font color="Green" title="E22">A30</font></i>: 
( <font color="Maroon" title="c13">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#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font></span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font> holds <br/><font color="Maroon" title="c13">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</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="E11:10_5_2"/>
( ( for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c13">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> ) &amp;  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:10_5"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="zfmodel1.html#E10:10_5_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="zf_lang.html#D39" title="ZF_LANG:def.39">ZF_LANG:def 39</a></i>;<br/><a NAME="E12:10_5_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c13">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10_5"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="zf_lang.html#T81" title="ZF_LANG:th.81">ZF_LANG:81</a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><a NAME="E13:10_5_2"/><b>then </b><i><font color="Green" title="E23">A31</font></i>: 
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c13">m</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> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E4:10_5"><i><font color="Green" title="E16">A24</font></i></a>, <a class="txt" href="zfmodel1.html#E1:10_5_2"><i><font color="Green" title="E17">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:10_5_2_1"/><i><font color="Green" title="E24">A32</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>
 ;<br/><a NAME="E2:10_5_2_1"/><b>then </b><i><font color="Green" title="E25">A33</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:10_5_2"><i><font color="Green" title="E22">A30</font></i></a></i>;<br/><b>assume </b><a NAME="E3:10_5_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y</font>
 ;<br/><a NAME="E4:10_5_2_1"/><b>then </b>
( <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> &amp; <font color="Maroon" title="c13">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> &amp; <font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:10_5_2"><i><font color="Green" title="E18">A26</font></i></a>, <a class="txt" href="zfmodel1.html#E7:10_5_2"><i><font color="Green" title="E20">A28</font></i></a>, <a class="txt" href="zfmodel1.html#E10:10_5_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="txt" href="zfmodel1.html#E2:10_5_2_1"><i><font color="Green" title="E25">A33</font></i></a></i>;<br/><b>hence </b><a NAME="E5:10_5_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:10_5_2_1"><i><font color="Green" title="E24">A32</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:10_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E8:10_5_2"><i><font color="Green" title="E21">A29</font></i></a>, <a class="txt" href="zfmodel1.html#E13:10_5_2"><i><font color="Green" title="E23">A31</font></i></a>, <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="E6:10_5"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:10_5"><i><font color="Green" title="E15">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:10"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="zfmodel1.html#E5:10"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="zfmodel1.html#E6:10"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="zfmodel1.html#E7:10"><i><font color="Green" title="E12">A16</font></i></a>, <a class="ref" href="zf_lang.html#T25" title="ZF_LANG:th.25">ZF_LANG:25</a></i>;<br/>


</div>
