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

<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>thus </b><a NAME="E1:105"/>
( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> implies  <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1"/>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>
 ;<br/>

<a NAME="E2:105_1"/><b>then </b>
( <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a>  or <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a>  )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D15" target="_self" title="ZF_LANG:def.15">Def15</a></i>;<br/>
<a NAME="E3:105_1"/><b>then </b>
( <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c1">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="c1">H</font></span>)</span> or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c1">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="c1">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T53" target="_self" title="ZF_LANG:th.53">Th53</a>, <a class="ref" href="zf_lang.html#T54" target="_self" title="ZF_LANG:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1"/>
 <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T102" target="_self" title="ZF_LANG:th.102">Th102</a>, <a class="ref" href="zf_lang.html#T103" target="_self" title="ZF_LANG:th.103">Th103</a></i>;<br/>


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

<b>assume </b><a NAME="E2:105"/><i><font color="Green" title="E85">A1</font></i>: 
 <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E3:105"/>
not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>
 ;<br/>

<a NAME="E4:105"/><b>then </b>
( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self" title="ZF_LANG:th.26">Th26</a></i>;<br/>
<a NAME="E5:105"/><b>then </b><i><font color="Green" title="E86">A2</font></i>: 
( <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">H</font></span>)</span> or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">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="c1">H</font></span>)</span> or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN: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#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c1">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="c1">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30" target="_self" title="ZF_LANG:def.30">Def30</a>, <a class="ref" href="zf_lang.html#T58" target="_self" title="ZF_LANG:th.58">Th58</a>, <a class="ref" href="zf_lang.html#T62" target="_self" title="ZF_LANG:th.62">Th62</a></i>;<br/>
<div><i><font color="Green" title="E87">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">H</font></span>)</span>
 ;<br/><a NAME="E2:105_2"/><b>then </b><i><font color="Green" title="E88">A4</font></i>: 
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T71" target="_self" title="ZF_LANG:th.71">Th71</a></i>;<br/><a NAME="E3:105_2"/><b>then </b>
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self" title="ZF_LANG:th.82">Th82</a></i>;<br/><a NAME="E4:105_2"/><b>then </b>
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></i>;<br/><a NAME="E5:105_2"/><b>then </b><i><font color="Green" title="E89">A5</font></i>: 
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></i>;<br/><a NAME="E6:105_2"/>
 <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="c1">H</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105_2"><i><font color="Green" title="E88">A4</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/><b>hence </b><a NAME="E7:105_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="zf_lang.html#E5:105_2"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_3"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">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="c1">H</font></span>)</span>
 ;<br/><a NAME="E2:105_3"/><b>then </b><i><font color="Green" title="E88">A6</font></i>: 
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T72" target="_self" title="ZF_LANG:th.72">Th72</a></i>;<br/><a NAME="E3:105_3"/><b>then </b>
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self" title="ZF_LANG:th.82">Th82</a></i>;<br/><a NAME="E4:105_3"/><b>then </b>
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></i>;<br/><a NAME="E5:105_3"/><b>then </b><i><font color="Green" title="E89">A7</font></i>: 
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></i>;<br/><a NAME="E6:105_3"/>
(  <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="c1">H</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c1">H</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105_3"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/><b>hence </b><a NAME="E7:105_3"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="zf_lang.html#E5:105_3"><i><font color="Green" title="E89">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:105"/><b>then </b><i><font color="Green" title="E88">A8</font></i>: 
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="zf_lang.html#E5:105"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="zf_lang.html#E6:105"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="zf_lang.html#T73" target="_self" title="ZF_LANG:th.73">Th73</a></i>;<br/>
<a NAME="E9:105"/><b>then </b>
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self" title="ZF_LANG:th.82">Th82</a></i>;<br/>
<a NAME="E10:105"/><b>then </b>
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></i>;<br/>
<a NAME="E11:105"/><b>then </b><i><font color="Green" title="E89">A9</font></i>: 
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K29" title="ZF_LANG:func.29">Subformulae</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></i>;<br/>
<a NAME="E12:105"/>
 <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="c1">H</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="zf_lang.html#E8:105"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E13:105"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="zf_lang.html#E11:105"><i><font color="Green" title="E89">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
