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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>thus </b><a NAME="E1:105"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V7" target="_self">atomic</a> implies  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</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">c<sub>1</sub></font> is <a href="zf_lang.html#V7" target="_self">atomic</a>
 ;<br/>

<a NAME="E2:105_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:105_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" target="_self">Var1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zf_lang.html#K4" target="_self">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" target="_self">Var2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" target="_self">Var1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zf_lang.html#K5" target="_self">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" target="_self">Var2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T53" target="_self">Th53</a>, <a class="ref" href="zf_lang.html#T54" target="_self">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1"/>
 <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T102" target="_self">Th102</a>, <a class="ref" href="zf_lang.html#T103" target="_self">Th103</a></i>;<br/>


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

<b>assume </b><a NAME="E2:105"/><i><font color="Green">E85</font></i>: 
 <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E3:105"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V7" target="_self">atomic</a>
 ;<br/>

<a NAME="E4:105"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V4" target="_self">negative</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V6" target="_self">universal</a> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E5:105"/><b>then </b><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8" target="_self">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" target="_self">bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30" target="_self">Def30</a>, <a class="ref" href="zf_lang.html#T58" target="_self">Th58</a>, <a class="ref" href="zf_lang.html#T62" target="_self">Th62</a></i>;<br/>
<div><i><font color="Green">E87</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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:105_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E3:105_2"/><b>then </b>
 <a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self">Th82</a></i>;<br/><a NAME="E4:105_2"/><b>then </b>
 <a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self">Def41</a></i>;<br/><a NAME="E5:105_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self">Def42</a></i>;<br/><a NAME="E6:105_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self">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">E85</font></i></a>, <a class="txt" href="zf_lang.html#E5:105_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tarski.html#D1">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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:105_3"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T72" target="_self">Th72</a></i>;<br/><a NAME="E3:105_3"/><b>then </b>
(  <a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self">Th82</a></i>;<br/><a NAME="E4:105_3"/><b>then </b>
(  <a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self">Def41</a></i>;<br/><a NAME="E5:105_3"/><b>then </b><i><font color="Green">E89</font></i>: 
(  <a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self">Def42</a></i>;<br/><a NAME="E6:105_3"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:105_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self">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">E85</font></i></a>, <a class="txt" href="zf_lang.html#E5:105_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:105"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang.html#E5:105"><i><font color="Green">E86</font></i></a>, <a class="txt" href="zf_lang.html#E6:105"><i><font color="Green">E87</font></i></a>, <a class="ref" href="zf_lang.html#T73" target="_self">Th73</a></i>;<br/>
<a NAME="E9:105"/><b>then </b>
 <a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E10:105"/><b>then </b>
 <a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D41" target="_self">Def41</a></i>;<br/>
<a NAME="E11:105"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D42" target="_self">Def42</a></i>;<br/>
<a NAME="E12:105"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang.html#E8:105"><i><font color="Green">E88</font></i></a>, <a class="ref" href="zf_lang.html#T81" target="_self">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">E85</font></i></a>, <a class="txt" href="zf_lang.html#E11:105"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
