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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">H</font>) iff <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">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="c2">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) iff <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:70"/>
( <font color="Maroon" title="c2">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) implies <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> )
  <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="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c2">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E57">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c4">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
(  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">G</font> or  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">F</font> )
 ; <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="E3:70_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" target="_self" title="ZF_LANG:th.16">Th33</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="zf_lang.html#T17" target="_self" title="ZF_LANG:th.17">Th34</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="E58">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:70_1_2"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">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> <font color="Maroon" title="c2">F</font>
 ; <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="E2:70_1_2"/>
<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T17" target="_self" title="ZF_LANG:th.17">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</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="E3:70_1"/>
<font color="Maroon" title="c2">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>)
 ; <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="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font></span><br/>

<a NAME="E4:70_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">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> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="zf_lang.html#D39" target="_self" title="ZF_LANG:def.39">Def39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:70_1"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T3" target="_self" title="ZF_LANG:th.3">Th12</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">thus </span><a NAME="E2:70"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c2">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">H</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D39" target="_self" title="ZF_LANG:def.39">Def39</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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