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

<span class="kw">let </span><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  iff  <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> )</span><br/>

<span class="kw">thus </span><a NAME="E1:104"/>
( <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> implies <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:104_1"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/>

<a NAME="E2:104_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  or <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D15" target="_self" title="ZF_LANG:def.15">Def15</a></span>;<br/>
<a NAME="E3:104_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T36" target="_self" title="ZF_LANG:th.36">Th53</a>, <a class="ref" href="zf_lang.html#T37" target="_self" title="ZF_LANG:th.37">Th54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:104_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T80" target="_self" title="ZF_LANG:th.80">Th102</a>, <a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th103</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:104"/><span class="lab"><font color="Green" title="E83">A1</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> </span><br/>

<div><span class="lab"><font color="Green" title="E84">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:104_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A3</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T52" target="_self" title="ZF_LANG:th.52">Th71</a></span>;<br/><a NAME="E3:104_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T61" target="_self" title="ZF_LANG:th.61">Th82</a></span>;<br/><a NAME="E4:104_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></span>;<br/><a NAME="E5:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></span>;<br/><a NAME="E6:104_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="ref" href="zf_lang.html#T60" target="_self" title="ZF_LANG:th.60">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E7:104_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E5:104_2"><span class="lab"><font color="Green" title="E86">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E85">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:104_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A6</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T53" target="_self" title="ZF_LANG:th.53">Th72</a></span>;<br/><a NAME="E3:104_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T61" target="_self" title="ZF_LANG:th.61">Th82</a></span>;<br/><a NAME="E4:104_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></span>;<br/><a NAME="E5:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A7</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></span>;<br/><a NAME="E6:104_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_3"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="ref" href="zf_lang.html#T60" target="_self" title="ZF_LANG:th.60">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E7:104_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E5:104_3"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E5:104"/>
 not <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:104"/><span class="kw">then </span>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T10" target="_self" title="ZF_LANG:th.10">Th26</a></span>;<br/>
<a NAME="E7:104"/><span class="kw">then </span>
( <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>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D30" target="_self" title="ZF_LANG:def.30">Def30</a>, <a class="ref" href="zf_lang.html#T40" target="_self" title="ZF_LANG:th.40">Th58</a>, <a class="ref" href="zf_lang.html#T44" target="_self" title="ZF_LANG:th.44">Th62</a></span>;<br/>
<a NAME="E8:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A8</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="zf_lang.html#T54" target="_self" title="ZF_LANG:th.54">Th73</a></span>;<br/>
<a NAME="E9:104"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T61" target="_self" title="ZF_LANG:th.61">Th82</a></span>;<br/>
<a NAME="E10:104"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D41" target="_self" title="ZF_LANG:def.41">Def41</a></span>;<br/>
<a NAME="E11:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A9</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D42" target="_self" title="ZF_LANG:def.42">Def42</a></span>;<br/>
<a NAME="E12:104"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:104"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="ref" href="zf_lang.html#T60" target="_self" title="ZF_LANG:th.60">Th81</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:104"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E11:104"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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